{"id":5454,"date":"2024-11-18T11:01:47","date_gmt":"2024-11-18T03:01:47","guid":{"rendered":"http:\/\/jhwk.online\/?p=3661"},"modified":"2025-12-24T14:52:13","modified_gmt":"2025-12-24T06:52:13","slug":"%e7%ab%8b%e4%bd%93%e5%87%a0%e4%bd%95%e4%b8%8e%e7%a9%ba%e9%97%b4%e5%90%91%e9%87%8f-2","status":"publish","type":"post","link":"http:\/\/jhwk.online\/?p=5454","title":{"rendered":"\u7acb\u4f53\u51e0\u4f55"},"content":{"rendered":"\n<div class=\"wp-block-ht-block-toc  is-style-outline htoc htoc--position-wide toc-list-style-plain\" data-htoc-state=\"expanded\" data-htoc-scrollspy=\"false\" data-htoc-auto-collapse=\"true\" data-htoc-mapping-headers=\"[false,true,true,true,true,true]\"><span class=\"htoc__title\"><span class=\"ht_toc_title\">\u76ee\u5f55<\/span><span class=\"htoc__toggle\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\"><g fill=\"#444\"><path d=\"M15 7H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1z\"><\/path><path d=\"M15 1H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1zM15 13H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1z\"><\/path><\/g><\/svg><\/span><\/span><div class=\"htoc__itemswrap\"><ul class=\"ht_toc_list\"><li class=\"\"><a href=\"#htoc-\">\u6c42\u51e0\u4f55\u4f53\u7684\u8868\u9762\u79ef\u3001\u4f53\u79ef<\/a><\/li><li class=\"\"><a href=\"#htoc-11111\">\u4e0e\u7403\u6709\u5173\u7684\u5207\u3001\u63a5\u95ee\u9898<\/a><\/li><li class=\"\"><a href=\"#htoc-11111111111111111111111111\">\u5e73\u884c\u4e0e\u5782\u76f4<\/a><\/li><li class=\"\"><a href=\"#htoc-1111111111111111111111111111111111111111\">\u4ea4\u7ebf\u53ca\u622a\u9762\u95ee\u9898<\/a><\/li><\/ul><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-\">\u6c42\u51e0\u4f55\u4f53\u7684\u8868\u9762\u79ef\u3001\u4f53\u79ef<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.(1)\u5982\u679c\u4e00\u4e2a\u6c34\u5e73\u653e\u7f6e\u7684\u56fe\u5f62\u7684\u659c\u4e8c\u6d4b\u76f4\u89c2\u56fe\u662f\u4e00\u4e2a\u5e95\u89d2\u4e3a  45^{\\circ} , \u8170\u548c\u4e0a\u5e95\u5747\u4e3a 1\u7684~~~~\\\\\u7b49\u8170\u68af\u5f62, \u6c42\u539f\u5e73\u9762\u56fe\u5f62\u7684\u9762\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-2-sqrt-2-frac-s-s-frac-sqrt-2-4\"><pre>\u7b54\u6848: 2+\\sqrt{2} ;\u63d0\u793a:\u6cd5\u4e00\u3001\u753b\u51fa\u539f\u56fe;\u6cd5\u4e8c\u3001\u5229\u7528\u5173\u7cfb\\frac{S_{\u76f4\u89c2\u56fe}}{S_{\u539f\u56fe}} =\\frac{\\sqrt{2} }{4} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(2)\u5df2\u77e5\u67d0\u5706\u67f1\u4e0e\u5706\u9525\u7684\u9ad8\u76f8\u7b49,\u5b83\u4eec\u7684\u4f53\u79ef\u4e4b\u6bd4\u7b49\u4e8e\u4fa7\u9762\u79ef\u4e4b\u6bd4\u7684\u5e73\u65b9,\u5219\u5706\u67f1\u4e0e\u5706~\\\\\u9525\u7684\u6bcd\u7ebf\u957f\u4e4b\u6bd4\u4e3a(\\quad)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA. \\frac{\\sqrt{3}}{2}\\quad B. \\frac{\\sqrt{6}}{4}\\quad C. \\frac{\\sqrt{3}}{3}\\quad D. \\frac{1}{2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-da11\"><pre>\u7b54\u6848:A;\u63d0\u793a:\\frac{\\pi r_{1}^{2}\\cdot h}{\\displaystyle \\frac{1}{3}\\pi r_{2}^{2}h}=(\\frac{2\\pi r_{1}h}{\\pi r_{2}\\sqrt{h^{2}+r_{2}^{2}}})^{2}\\Rightarrow h = \\sqrt{3}r_{2}\\Rightarrow\\frac{h}{l}=\\frac{\\sqrt{3}}{2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(3)\u73b0\u6709\u4e00\u4e2a\u5706\u9525\u4e0e\u4e00\u4e2a\u7403,\u5b83\u4eec\u7684\u8868\u9762\u79ef\u76f8\u7b49,\u5706\u9525\u7684\u6bcd\u7ebf\u957f\u4e0e\u7403\u7684\u76f4\u5f84\u76f8\u7b49,\u6c42\u5706~~~\\\\\u9525\u7684\u5e95\u9762\u76f4\u5f84\u4e0e\u6bcd\u7ebf\u957f\u7684\u6bd4\u503c.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-da111\"><pre>\u7b54\u6848:\\sqrt{5}-1;\u63d0\u793a:\u8bbe\u5706\u9525\u7684\u5e95\u9762\u534a\u5f84\u4e3ar,\u6bcd\u7ebf\u957f\u4e3al,\u5219\u5176\u8868\u9762\u79ef\u4e3a\\pi rl+\\pi r^{2},~~~~~~~~~\\\\\u7403\u7684\u8868\u9762\u79ef\u4e3a4 \\pi(\\frac{l}{2})^{2}=\\pi l^{2},\u6240\u4ee5\\pi rl+\\pi r^{2}=\\pi l^{2},\u5373(\\frac{r}{l})^{2}+\\frac{r}{l}-1=0,~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u89e3\u5f97\\frac{r}{l}=\\frac{\\sqrt{5}-1}{2}\uff08\u8d1f\u503c\u820d\u53bb\uff09,\u6545\u5706\u9525\u7684\u5e95\u9762\u76f4\u5f84\u4e0e\u6bcd\u7ebf\u957f\u7684\u6bd4\u503c\\frac{2r}{l}=\\sqrt{5}-1.~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2. (1)\u5982\u56fe\u6240\u793a\u4e94\u9762\u4f53, \u5176\u4e2d\u56db\u8fb9\u5f62  A B C D  \u662f\u8fb9\u957f\u4e3a 2 \u7684\u6b63\u65b9\u5f62, \u4e14  \\triangle A D E, \\triangle B C F  \u5747\u4e3a\u6b63\\\\\u4e09\u89d2\u5f62,  E F \/ \/ C D, E F=4 , \u6c42\u8be5\u51e0\u4f55\u4f53\u7684\u4f53\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-bfc93e5\" data-block-id=\"bfc93e5\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-bfc93e5-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-8462894\" data-v=\"4\" data-block-id=\"8462894\"><style>@media screen and (min-width:690px){.stk-8462894 {flex:var(--stk-flex-grow, 1) 1 calc(75\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-frac-8-sqrt-2-3\"><pre>\u7b54\u6848:\\frac{8 \\sqrt{2}}{3} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-5cc753d\" data-block-id=\"5cc753d\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-5cc753d-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-18affd2\" data-v=\"4\" data-block-id=\"18affd2\"><style>@media screen and (min-width:690px){.stk-18affd2 {flex:var(--stk-flex-grow, 1) 1 calc(66.667\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-v-v-e-a-d-g-v-f-bhc-v-a-d-g-b-c-h-frac-8-sqrt-2-3\"><pre>\u6cd5\u4e00\u3001\u5207\u5272\u5982\u56fe\uff0cV=V_{ E-A D G}+~~~~\\\\+V_{ F-BHC}+V_{  \u67f1  A D G-B C H}= \\frac{8 \\sqrt{2}}{3}.~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-b3f333b\" data-v=\"4\" data-block-id=\"b3f333b\"><style>@media screen and (min-width:690px){.stk-b3f333b {flex:var(--stk-flex-grow, 1) 1 calc(33.333\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/02\/ltjh1-1.png\" alt=\"\" class=\"wp-image-4501\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-f36b9df\" data-block-id=\"f36b9df\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-f36b9df-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-3bc9e8b\" data-v=\"4\" data-block-id=\"3bc9e8b\"><style>@media screen and (min-width:690px){.stk-3bc9e8b {flex:var(--stk-flex-grow, 1) 1 calc(66.667\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-frac-1-2-v-a-bcd\"><pre>\u6cd5\u4e8c\u3001\u8865\u6210\u6b63\u56db\u9762\u4f53\uff0c\u6240\u6c42\u4f53\u79ef\u4e3a~~~\\\\\\frac{1}{2} V_{A-BCD}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f11a6da\" data-v=\"4\" data-block-id=\"f11a6da\"><style>@media screen and (min-width:690px){.stk-f11a6da {flex:var(--stk-flex-grow, 1) 1 calc(33.333\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/02\/ltjh1-1-1-1.png\" alt=\"\" class=\"wp-image-4502\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-3c4631e\" data-v=\"4\" data-block-id=\"3c4631e\"><style>@media screen and (min-width:690px){.stk-3c4631e {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"664\" height=\"543\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh1.png\" alt=\"\" class=\"wp-image-3966\" style=\"width:139px;height:auto\" srcset=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh1.png 664w, http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh1-300x245.png 300w\" sizes=\"auto, (max-width: 664px) 100vw, 664px\" \/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(2)\u5728\u6b63\u56db\u68f1\u53f0  A_{1} B_{1} C_{1} D_{1}-A B C D  \u4e2d,  A A_{1} \/\/  \u5e73\u9762  C_{1} B D, A B=2 A A_{1}=4 , \u6c42\u6b63\u56db\u68f1\u53f0\\\\ A_{1} B_{1} C_{l} D_{1}-A B C D  \u7684\u4f53\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-7ca1206\" data-block-id=\"7ca1206\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-7ca1206-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-a53e2ec\" data-v=\"4\" data-block-id=\"a53e2ec\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-a53e2ec-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-a53e2ec-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-0384195\" data-block-id=\"0384195\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-0384195-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-180afed\" data-v=\"4\" data-block-id=\"180afed\"><style>@media screen and (min-width:690px){.stk-180afed {flex:var(--stk-flex-grow, 1) 1 66.7\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d111111111111\"><pre>\u7b54\u6848:\\frac{28\\sqrt{2}}{3};\u63d0\u793a:\u5982\u56fe\uff0cA_1C_1=\\frac{1}{2}AC=2\\sqrt{2},~~~~~~~~\\\\\u68f1\u53f0\u9ad8A_1M=\\sqrt{2},V=\\frac{28\\sqrt{2}}{3}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-9df4b4f\" data-v=\"4\" data-block-id=\"9df4b4f\"><style>@media screen and (min-width:690px){.stk-9df4b4f {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/02\/ltjh2-2.png\" alt=\"\" class=\"wp-image-4430\" style=\"width:211px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n<\/details>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3. \u5982\u56fe, \u5728\u5e73\u9762\u4e94\u8fb9\u5f62  A B C D E  \u4e2d,  A B=D E=1, B C=C D=2, A E=\\sqrt{2}, \\angle A B C   =~\\\\\\angle B C D=\\angle C D E=90^{\\circ} ,\u6c42\u4e94\u8fb9\u5f62  A B C D E  \u7ed5\u76f4\u7ebf  A B  \u65cb\u8f6c\u4e00\u5468\u6240\u6210\u7684\u51e0\u4f55\u4f53\u7684\u4f53\u79ef .~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-07c121a\" data-block-id=\"07c121a\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-07c121a-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f2f04d6\" data-v=\"4\" data-block-id=\"f2f04d6\"><style>@media screen and (min-width:690px){.stk-f2f04d6 {flex:var(--stk-flex-grow, 1) 1 calc(75\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-frac-23-pi-3-v-v-text-v-text-frac-23-pi-3\"><pre>\u7b54\u6848:  \\frac{23 \\pi}{3} ; \u63d0\u793a:  V=V_{\\text {\u5706\u67f1 }}-V_{\\text {\u5706\u9525}}=   \\frac{23 \\pi}{3} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-2855f8c\" data-v=\"4\" data-block-id=\"2855f8c\"><style>@media screen and (min-width:690px){.stk-2855f8c {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh2.png\" alt=\"\" style=\"width:153px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4. \u5982\u56fe, \u56db\u8fb9\u5f62  A B C D  \u4e3a\u77e9\u5f62,  P A \\perp  \u5e73\u9762  A B C D, P A \/ \/ B E, P A=B C=2 B E=2 A B \\\\  =2 , \u5206\u522b\u6c42\u4e09\u68f1\u9525  E-P B C, E-P A C, P-C D E  \u7684\u4f53\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-aed6311\" data-block-id=\"aed6311\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-aed6311-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-46b985c\" data-v=\"4\" data-block-id=\"46b985c\"><style>@media screen and (min-width:690px){.stk-46b985c {flex:var(--stk-flex-grow, 1) 1 calc(75\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-a4206ae\" data-block-id=\"a4206ae\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-a4206ae-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-a694037\" data-v=\"4\" data-block-id=\"a694037\"><style>@media screen and (min-width:690px){.stk-a694037 {flex:var(--stk-flex-grow, 1) 1 calc(66.667\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-frac-1-3-frac-2-3-frac-1-3-v-e-p-b-c-v-c-p-b-e-frac-1-3-v-e-p-a-c-v-c-pea-frac-2-3-v-p-cde-v-p-dfe-v-d-pfe-frac-1-3\"><pre>\u7b54\u6848:\\frac{1}{3},\\frac{2}{3},\\frac{1}{3};~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u63d0\u793a:V_{E-P B C}=V_{C-P B E}=\\frac{1}{3} ;~~~~~~~~~~\\\\\nV_{E-P A C}=V_{C-PEA}=\\frac{2}{3};~~~~~~~~~~~~~~~~~~~~~\\\\\nV_{P-CDE}=V_{P-DFE}=V_{D-PFE}=\\frac{1}{3} ;~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-b59afb6\" data-v=\"4\" data-block-id=\"b59afb6\"><style>@media screen and (min-width:690px){.stk-b59afb6 {flex:var(--stk-flex-grow, 1) 1 calc(33.333\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjh4-1.png\" alt=\"\" class=\"wp-image-4873\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-3ac4a55\" data-v=\"4\" data-block-id=\"3ac4a55\"><style>@media screen and (min-width:690px){.stk-3ac4a55 {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh4.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5. \u5982\u56fe, \u5728\u4e09\u68f1\u9525  P-A B C  \u4e2d,  A B \\perp B C, A B=2, B C=2 \\sqrt{2}, P B=P C=\\sqrt{6}, A C ,  ~~~~\\\\B C  \u7684\u4e2d\u70b9\u5206\u522b\u4e3a  F, O , \u82e5  \\angle P O F=120^{\\circ} , \u6c42\u4e09\u68f1\u9525  P-A B C  \u7684\u4f53\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-244dc24\" data-block-id=\"244dc24\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-244dc24-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-960c9a9\" data-v=\"4\" data-block-id=\"960c9a9\"><style>@media screen and (min-width:690px){.stk-960c9a9 {flex:var(--stk-flex-grow, 1) 1 calc(71.4\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-f-o-perp-b-c-p-o-perp-b-c-angle-p-o-f-p-b-c-a-p-p-m-perp-a-b-c-m-m-f-o-therefore-angle-p-o-m-60-circ-quad-rightarrow-p-m-sqrt-3-p-a-b-c-v-p-a-b-c-frac-1-3-s-triangle-a-b-c-cdot-p-m-frac-2-sqrt-6-3-because-f-o-perp-b-c-p-o-perp-b-c-v-c-p-o-f-frac-1-3-s-triangle-p-o-f-cdot-o-c-frac-sqrt-6-6-because-v-c-p-o-f-v-p-c-o-f-frac-1-4-v-p-a-b-c-therefore-v-p-a-b-c-frac-2-sqrt-6-3\"><pre>\u6cd5\u4e00\u3001\u6613\u77e5  F O \\perp B C, P O \\perp B C, \\angle P O F  \u662f\u4e8c\u9762\u89d2~~~~~~~~~~~~~ \\\\ P-B C-A  \u7684\u5e73\u9762\u89d2,\\angle P O F =120^\\circ,\u8fc7\u70b9  P  \u4f5c  P M \\perp  ~~~~~~~~\\\\\u5e73\u9762  A B C  \u4e8e\u70b9  M , \u5219  M  \u5728  F O  \u7684\u5ef6\u957f\u7ebf\u4e0a,\n\n\\therefore \\angle P O M=60^{\\circ}, \\\\ \\Rightarrow P M=\\sqrt{3} .\n\n\n\\therefore \n V_{P-A B C}=\\frac{1}{3} S_{\\triangle A B C} \\cdot P M=\\frac{2 \\sqrt{6}}{3} .~~~~~~~~~~~~~\\\\\n\u6cd5\u4e8c\u3001  \\because F O \\perp B C, P O \\perp B C ,\n V_{C-P O F}=\\frac{1}{3} S_{\\triangle P O F} \\cdot O C~~\\\\\n=\\frac{\\sqrt{6}}{6} ,\n\u53c8  \\because V_{C-P O F}=V_{P-C O F}=\\frac{1}{4} V_{P-A B C} ,~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\n\\therefore V_{P-A B C}=\\frac{2 \\sqrt{6}}{3}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-71e5dbf\" data-v=\"4\" data-block-id=\"71e5dbf\"><style>@media screen and (min-width:690px){.stk-71e5dbf {flex:var(--stk-flex-grow, 1) 1 calc(28.6\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhkj1.png\" alt=\"\"\/><\/figure>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh1-7.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>6. \u5982\u56fe, \u5728\u56db\u68f1\u96c9  P-A B C D  \u4e2d, \u56db\u8fb9\u5f62  A B C D  \u662f\u8fb9\u957f\u4e3a 2 \u7684\u6b63\u65b9\u5f62,  \\triangle P B C  \u4e3a\u6b63\u4e09\u89d2\u5f62\uff0c \\\\ M, N  \u5206\u522b\u4e3a  P D, B C  \u7684\u4e2d\u70b9\uff0c  P N \\perp A B ,\u6c42\u4e09\u68f1\u9525  P-A M N  \u7684\u4f53\u79ef;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-26302a1\" data-block-id=\"26302a1\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-26302a1-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-4a619a6\" data-v=\"4\" data-block-id=\"4a619a6\"><style>@media screen and (min-width:690px){.stk-4a619a6 {flex:var(--stk-flex-grow, 1) 1 66.7\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-v-p-a-m-n-v-a-p-i-n-frac-1-2-v-a-p-d-n-frac-1-2-v-p-a-d-n-frac-1-4-v-p-a-b-c-d-frac-sqrt-3-3\"><pre>\u63d0\u793a:V_{P-A M N}=V_{A-P I N}=\\frac{1}{2} V_{A-P D N}=\\frac{1}{2} V_{P-A D N}=\\\\\\frac{1}{4} V_{P-A B C D}=\\frac{\\sqrt{3}}{3}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-fdc7cee\" data-v=\"4\" data-block-id=\"fdc7cee\"><style>@media screen and (min-width:690px){.stk-fdc7cee {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhkj2.png\" alt=\"\" style=\"width:215px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-palette-color-2-color has-alpha-channel-opacity has-palette-color-2-background-color has-background is-style-wide\"\/>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-11111\">\u4e0e\u7403\u6709\u5173\u7684\u5207\u3001\u63a5\u95ee\u9898<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.\u6839\u636e\u6761\u4ef6\u6c42\u51e0\u4f55\u4f53\u7684\u5916\u63a5\u7403\u7684\u534a\u5f84.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-4d8b9fe\" data-block-id=\"4d8b9fe\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-4d8b9fe-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-ab20427\" data-v=\"4\" data-block-id=\"ab20427\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-ab20427-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-ab20427-inner-blocks\">\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(1)\u6761\u4ef6:PC\\perp \u9762ABC,\\\\AC\\perp AB,~~~~~~~~~~~~~~~~~~~~~\\\\PC=AC=AB=1.~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq1.png\" alt=\"\" style=\"width:162px;height:auto\"\/><\/figure>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-r-frac-sqrt-3-2\"><pre>\u7b54\u6848\uff1aR=\\frac{\\sqrt{3}}{2};~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq1-1.png\" alt=\"\" style=\"width:191px;height:auto\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f89ad17\" data-v=\"4\" data-block-id=\"f89ad17\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-f89ad17-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-f89ad17-inner-blocks\">\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(2)\u6761\u4ef6:PB\\perp \u9762ABC,\\\\AB\\perp BC,~~~~~~~~~~~~~~~~~~~~~\\\\PB=AB=BC=1.~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"428\" height=\"360\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq2.png\" alt=\"\" class=\"wp-image-3996\" style=\"aspect-ratio:1.1888715377660886;width:150px;height:auto\" srcset=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq2.png 428w, http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq2-300x252.png 300w\" sizes=\"auto, (max-width: 428px) 100vw, 428px\" \/><\/figure>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-r-frac-sqrt-3-21\"><pre>\u7b54\u6848\uff1aR=\\frac{\\sqrt{3}}{2};~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<p id=\"htoc-111111111111111111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"426\" height=\"366\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq2-2.png\" alt=\"\" class=\"wp-image-3997\" srcset=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq2-2.png 426w, http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq2-2-300x258.png 300w\" sizes=\"auto, (max-width: 426px) 100vw, 426px\" \/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-5622cef\" data-v=\"4\" data-block-id=\"5622cef\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-5622cef-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-5622cef-inner-blocks\">\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(3)\u6761\u4ef6:PA\\perp \u9762ABC,\\\\AC\\perp BC,~~~~~~~~~~~~~~~~~~~~~\\\\PA=AC=BC=1.~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq4.png\" alt=\"\" style=\"width:162px;height:auto\"\/><\/figure>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-r-frac-sqrt-3-211\"><pre>\u7b54\u6848\uff1aR=\\frac{\\sqrt{3}}{2};~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq4-1.png\" alt=\"\" style=\"width:191px;height:auto\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-palette-color-5-color has-alpha-channel-opacity has-palette-color-5-background-color has-background is-style-wide\"\/>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-66afde9\" data-block-id=\"66afde9\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-66afde9-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-a9d7441\" data-v=\"4\" data-block-id=\"a9d7441\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-a9d7441-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-a9d7441-inner-blocks\">\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(4)\u6761\u4ef6:\u5982\u56fe\u6b63\u56db\u9762\u4f53\\\\ABCD\uff0c\u68f1\u957f\u4e3a\\sqrt{2}.~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-spacer stk-block-spacer stk--no-padding stk-block stk-2bd4cf4\" data-block-id=\"2bd4cf4\"><style>.stk-2bd4cf4 {height:50px !important;}<\/style><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq3.png\" alt=\"\" style=\"width:188px;height:auto\"\/><\/figure>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-r-frac-sqrt-3-2111\"><pre>\u7b54\u6848\uff1aR=\\frac{\\sqrt{3}}{2};~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq3-1.png\" alt=\"\" style=\"width:191px;height:auto\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f28b2ed\" data-v=\"4\" data-block-id=\"f28b2ed\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-f28b2ed-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-f28b2ed-inner-blocks\">\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(5)\u6761\u4ef6:\u5bf9\u68f1\u76f8\u7b49~~~~~~~~\\\\AC=BD=\\sqrt{3},~~~~~~~~~~\\\\AD=BC=\\sqrt{5},~~~~~~~~~~\\\\AB=CD=\\sqrt{6}.~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq3-1.png\" alt=\"\" style=\"width:188px;height:auto\"\/><\/figure>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-r-frac-sqrt-5-2-a-b-c-left-begin-matrix-a-2-b-2-2-b-2-c-2-3-a-2-c-2-5-end-matrix-right-rightarrow-a-2-b-2-c-2-5\"><pre>\u7b54\u6848\uff1aR=\\frac{\\sqrt{5}}{2};~~~~~~~~~~~~\\\\\u63d0\u793a:\u8bbe\u957f\u3001\u5bbd\u3001\u9ad8\u5206~\\\\\u522b\u4e3aa,b,c\uff0c\u53ef\u5f97:~~~~~~~~~\\\\\\left\\{\\begin{matrix}\na^2+b^2=3 \\\\\nb^2+c^2=5\\\\\n\na^2+c^2=6\\end{matrix}\\right.\\\\\\Rightarrow a^2+b^2+c^2=7<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq3-1.png\" alt=\"\" style=\"width:194px;height:auto\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-b92f367\" data-v=\"4\" data-block-id=\"b92f367\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-b92f367-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-b92f367-inner-blocks\">\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(6)\u6761\u4ef6:\u5982\u56fe\u6b63\u4e09\u68f1\u9525,\\\\PA=PB=PC=4\\sqrt{5}\\\\AB=AC=BC=4\\sqrt{3}.<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq10.png\" alt=\"\" style=\"width:157px;height:auto\"\/><\/figure>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-r-5\"><pre>\u7b54\u6848\uff1aR=5;~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq10-1.png\" alt=\"\" style=\"width:193px;height:auto\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-palette-color-5-color has-alpha-channel-opacity has-palette-color-5-background-color has-background is-style-wide\"\/>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-81b70b1\" data-block-id=\"81b70b1\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-81b70b1-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-496440a\" data-v=\"4\" data-block-id=\"496440a\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-496440a-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-496440a-inner-blocks\">\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(7)\u6761\u4ef6:\u6b63\u4e09\u68f1\u67f1\u6ee1\u8db3\\\\AB=2,AA'=4.~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq6-1.png\" alt=\"\" style=\"width:128px;height:auto\"\/><\/figure>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-r-frac-4-sqrt-3-3\"><pre>\u7b54\u6848\uff1aR=\\frac{4\\sqrt{3}}{3};~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq6.png\" alt=\"\" style=\"width:183px;height:auto\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-16b2cad\" data-v=\"4\" data-block-id=\"16b2cad\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-16b2cad-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-16b2cad-inner-blocks\">\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(8)\u6761\u4ef6:\u5706\u53f0\u7684\u4e0a\u4e0b\u5e95\\\\\u534a\u5f84\u5206\u522b\u4e3a3,4,\u5706\u53f0\u7684\\\\\u9ad8\u4e3a7.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq7.png\" alt=\"\" style=\"width:169px;height:auto\"\/><\/figure>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-r-51\"><pre>\u7b54\u6848\uff1aR=5~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq7-1.png\" alt=\"\" style=\"width:187px;height:auto\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-61818db\" data-v=\"4\" data-block-id=\"61818db\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-61818db-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-61818db-inner-blocks\">\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(9)\u6761\u4ef6:\u5706\u53f0\u7684\u4e0a\u4e0b\u5e95\\\\\u534a\u5f84\u5206\u522b\u4e3a3,4,\u5706\u53f0\u7684\\\\\u9ad8\u4e3a1.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq8.png\" alt=\"\" style=\"width:201px;height:auto\"\/><\/figure>\n\n\n\n<div class=\"wp-block-stackable-spacer stk-block-spacer stk--no-padding stk-block stk-9167da7\" data-block-id=\"9167da7\"><style>.stk-9167da7 {height:19px !important;}<\/style><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-r-511\"><pre>\u7b54\u6848\uff1aR=5~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/wjq8-1.png\" alt=\"\" style=\"width:187px;height:auto\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-palette-color-5-color has-alpha-channel-opacity has-palette-color-5-background-color has-background is-style-wide\"\/>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2.\u56db\u68f1\u9525  P-A B C D  \u7684\u9876\u70b9\u90fd\u5728\u7403  O  \u7684\u8868\u9762\u4e0a,  \\triangle P A D  \u662f\u7b49\u8fb9\u4e09\u89d2\u5f62, \u5e95\u9762  A B C D \u662f~~~~\\\\\u77e9\u5f62, \u5e73\u9762  P A D \\perp  \u5e73\u9762  A B C D , \u82e5  A B=2, B C=3 , \u6c42\u7403  O  \u7684\u8868\u9762\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-c235da6\" data-block-id=\"c235da6\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-c235da6-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-4a0b9c6\" data-v=\"4\" data-block-id=\"4a0b9c6\"><style>@media screen and (min-width:690px){.stk-4a0b9c6 {flex:var(--stk-flex-grow, 1) 1 66.7\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d\"><pre>\u7b54\u6848:16\\pi;\u63d0\u793a:\u8865\u6210\u4e09\u68f1\u67f1\u5982\u4e0b\u56fe.~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjh13-1.png\" alt=\"\" class=\"wp-image-4875\" style=\"width:199px;height:auto\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-54c0de4\" data-v=\"4\" data-block-id=\"54c0de4\"><style>@media screen and (min-width:690px){.stk-54c0de4 {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh13.png\" alt=\"\" style=\"width:206px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.\u5982\u56fe\u6240\u793a, \u5728\u4e09\u68f1\u9525  S-A B C  \u4e2d,  \\triangle A B C  \u4e0e  \\triangle S B C  \u90fd\u662f\u8fb9\u957f\u4e3a 1 \u7684\u6b63\u4e09\u89d2\u5f62,\u4e8c\u9762\u89d2~~~~ \\\\ S-B C-A  \u7684\u5927\u5c0f\u4e3a  \\frac{2 \\pi}{3} , \u82e5  S, A, B, C  \u56db\u70b9\u90fd\u5728\u7403  O  \u7684\u8868\u9762\u4e0a, \u6c42\u7403  O \u7684\u8868\u9762\u79ef.~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-1e69a06\" data-block-id=\"1e69a06\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-1e69a06-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-cf9b54e\" data-v=\"4\" data-block-id=\"cf9b54e\"><style>@media screen and (min-width:690px){.stk-cf9b54e {flex:var(--stk-flex-grow, 1) 1 calc(71.7\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-frac-7-pi-3-angle-a-d-s-frac-2-pi-3-e-f-triangle-s-b-c-triangle-a-b-c-e-f-o-o-a-rightarrow-r-o-a-b-d-frac-1-2-a-d-frac-sqrt-3-2-d-e-frac-1-3-a-d-frac-sqrt-3-6-a-e-frac-2-3-a-d-frac-sqrt-3-3-o-d-rt-triangle-o-d-e-angle-o-d-e-frac-pi-3-o-e-sqrt-3-d-e-frac-1-2-therefore-o-a-2-o-e-2-a-e-2-frac-7-12-therefore-s-4-pi-r-2-frac-7-pi-3\"><pre>\u7b54\u6848: \\frac{7 \\pi}{3};\u63d0\u793a: \u5982\u56fe\uff0c\n \\angle S D A=\\frac{2 \\pi}{3} ,\n\u70b9  E, F , \u4e3a  \\triangle S B C  ~~~\\\\\u548c  \\triangle A B C  \u7684\u91cd\u5fc3,\n\u8fc7\u70b9  E, F  \u5206\u522b\u4f5c\u4e24\u9762\u7684\u5782\u7ebf, \u5176\u4ea4\u70b9\u5373\u7403\\\\\u5fc3  O , \u8fde\u63a5  O A, \\Rightarrow R=O A ,\n\u7531\u9898\u610f\u77e5  B D=\\frac{1}{2}, A D=\\frac{\\sqrt{3}}{2}, ~\\\\D E=\\frac{1}{3} A D=\\frac{\\sqrt{3}}{6}, A E=\\frac{2}{3} A D=\\frac{\\sqrt{3}}{3} ,\n\u8fde\u63a5  O D , \u5728~~~~~~~~~~\\\\ Rt  \\triangle O D E  \u4e2d,  \\angle O D E=\\frac{\\pi}{3}, O E=\\sqrt{3} D E=\\frac{1}{2} ,~~~~~~~~~~~~~~~~~~~~\\\\\n \\therefore O A^{2}=O E^{2}+A E^{2}=\\frac{7}{12}, \\therefore  \u8868\u9762\u79ef\u4e3a  S=4 \\pi R^{2}=\\frac{7 \\pi}{3} .~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f707225\" data-v=\"4\" data-block-id=\"f707225\"><style>@media screen and (min-width:690px){.stk-f707225 {flex:var(--stk-flex-grow, 1) 1 calc(28.3\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/pmxl12-1-1024x736.png\" alt=\"\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-1111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"749\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/pmxl12-1-1-1024x749.png\" alt=\"\" class=\"wp-image-3993\" srcset=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/pmxl12-1-1-1024x749.png 1024w, http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/pmxl12-1-1-300x220.png 300w, http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/pmxl12-1-1-768x562.png 768w, http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/pmxl12-1-1.png 1204w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u53d8\u5f0f1.\u5c06\u4e0a\u9898\u4e2d\u7684\u4e8c\u9762\u89d2 S-B C-A  \u7684\u5927\u5c0f\u4e3a  \\displaystyle\\frac{\\pi}{2} ,  \u5176\u5b83\u6761\u4ef6\u4e0d\u53d8\uff0c\u6c42\u7403  O \u7684\u8868\u9762\u79ef.~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d11\"><pre>\u7b54\u6848:\\frac{5\\pi}{3};\u63d0\u793a:\u65b9\u6cd5\u5982\u4e0a\u9898,\u7403\u534a\u5f84R=\\frac{\\sqrt{15}}{6}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u53d8\u5f0f2. \u5728\u4e09\u68f1\u9525  S-A B C  \u4e2d, SC\\perp SB,AC \\perp AB,SC=AB=4,SB=AC=3,\u6c42\u4e09\\\\\u68f1\u9525  S-A B C \u5916\u63a5\u7403\u7684\u8868\u9762\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d1\"><pre>\u7b54\u6848:\\sqrt{5};\u63d0\u793a:\u659c\u8fb9BC\u7684\u4e2d\u70b9O\u4e3a\u7403\u5fc3\uff0c\u56e0\u4e3aO\u70b9\u5230\u6bcf\u4e2a\u9876\u70b9\u7684\u8ddd\u79bb\u76f8\u7b49\uff0c(\u6839\u636e\u76f4~\\\\\u89d2\u4e09\u89d2\u5f62\u659c\u8fb9\u7684\u4e2d\u70b9\u5230\u9876\u7684\u8ddd\u79bb\u76f8\u7b49\u53ef\u77e5).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u53d8\u5f0f3.\u5728\u4e09\u68f1\u9525  P-A B C  \u4e2d, \u5e73\u9762  P A C \\perp  \u5e73\u9762  A B C, P A=P C=A B=2 \\sqrt{3}, A C=4 ,  \\\\\\angle B A C=30^{\\circ} , \u6c42\u8be5\u4e09\u68f1\u9525\u5916\u63a5\u7403\u7684\u4f53\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-7e84101\" data-block-id=\"7e84101\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-7e84101-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-0cbd2cb\" data-v=\"4\" data-block-id=\"0cbd2cb\"><style>@media screen and (min-width:690px){.stk-0cbd2cb {flex:var(--stk-flex-grow, 1) 1 calc(75\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-9-sqrt-2-pi-o-p-a-c-o-a-o-a-o-p-r-p-o-1-2-sqrt-2-rt-triangle-o-o-1-a-r-2-2-sqrt-2-r-2-4-r-frac-3-sqrt-2-2-v-frac-4-3-pi-r-3-9-sqrt-2-pi\"><pre>\u7b54\u6848:  9 \\sqrt{2} \\pi ;\u63d0\u793a:\n\u7403\u5fc3\u4e3a  O  \u5728\u5e73\u9762  P A C  \u7684\u4e2d\u7ebf\u4e0a, \u8fde\u63a5  O A ,~\\\\ \u5219  O A=O P=R, P O_{1}=2 \\sqrt{2} .\u5728 Rt  \\triangle O O_{1} A  \u4e2d, \u5373 ~~~~~~~~~~~~~~~~~~\\\\ R^{2}=(2 \\sqrt{2}-R)^{2}+4 , \u5219  R=\\frac{3 \\sqrt{2}}{2} .  V=\\frac{4}{3} \\pi R^{3}=9 \\sqrt{2} \\pi .~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-4043b6a\" data-v=\"4\" data-block-id=\"4043b6a\"><style>@media screen and (min-width:690px){.stk-4043b6a {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh15.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4.\u5df2\u77e5\u5728\u56db\u9762\u4f53  V-A B C  \u4e2d,  V A=V B=V C=\\sqrt{3}, A B=\\sqrt{2}, \\angle A C B=\\frac{\\pi}{4} , \u6c42\u8be5\u56db\u9762\\\\\u4f53\u5916\u63a5\u7403\u7684\u8868\u9762\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-aa32ebc\" data-block-id=\"aa32ebc\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-aa32ebc-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-1879a9e\" data-v=\"4\" data-block-id=\"1879a9e\"><style>@media screen and (min-width:690px){.stk-1879a9e {flex:var(--stk-flex-grow, 1) 1 calc(70.5\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d1111\"><pre>\u7b54\u6848:\\frac{9 \\pi}{2} ;\u63d0\u793a:  V A=V B=V C=\\sqrt{3} \\therefore V  \u7531\u6b63\u5f26\u5b9a\u7406\\\\\u5f97:\n\u4e09\u89d2\u5f62  A B C  \u7684\u5916\u63a5\u5706\u7684\u534a\u5f84r=\\frac{\\sqrt{2}}{2 \\displaystyle \\sin \\frac{\\pi}{4}}=1 ;\n\u8bbe\u56db\u9762\\\\\u4f53\u5916\u63a5\u7403\u7684\u534a\u5f84\u4e3a  R,(\\sqrt{2}-R)^{2}=R^{2}-1 . \u89e3\u5f97: ~~~~~~~~~~~~~~\\\\ R=\\frac{3 \\sqrt{2}}{4} ,\n\u6240\u4ee5V=\\frac{9 \\pi}{2} ~.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-5de5b1e\" data-v=\"4\" data-block-id=\"5de5b1e\"><style>@media screen and (min-width:690px){.stk-5de5b1e {flex:var(--stk-flex-grow, 1) 1 calc(29.5\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"244\" height=\"235\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/pmxl1-11.png\" alt=\"\" class=\"wp-image-3861\" style=\"width:184px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5.\u534a\u7403\u5185\u653e\u4e09\u4e2a\u534a\u5f84\u4e3a\\sqrt{3}\u7684\u5c0f\u7403\uff0c\u4e09\u5c0f\u7403\u4e24\u4e24\u76f8\u5207\uff0c\u5e76\u4e14\u4e0e\u7403\u9762\u53ca\u534a\u7403\u5e95\u9762\u7684\u5927\u5706\u9762\\\\\u4e5f\u76f8\u5207\uff0c\u6c42\u8be5\u534a\u7403\u7684\u534a\u5f84.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-button-group stk-block-button-group stk-block stk-96da943\" data-block-id=\"96da943\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-button-group\">\n<div class=\"wp-block-stackable-button stk-block-button stk-block stk-4ec6b41\" data-block-id=\"4ec6b41\"><style>.stk-4ec6b41 {border-top-left-radius:13px !important;border-top-right-radius:13px !important;border-bottom-right-radius:13px !important;border-bottom-left-radius:13px !important;overflow:hidden !important;}<\/style><a class=\"stk-link stk-button stk--hover-effect-darken\" href=\"https:\/\/www.geogebra.org\/m\/fjf6nwpe\" target=\"_blank\" rel=\"noreferrer noopener\"><span class=\"stk-button__inner-text\">\u753b\u677f\u5b9e\u9a8c<\/span><\/a><\/div>\n<\/div><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-1d4d6bc\" data-block-id=\"1d4d6bc\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-1d4d6bc-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-46020c5\" data-v=\"4\" data-block-id=\"46020c5\"><style>@media screen and (min-width:690px){.stk-46020c5 {flex:var(--stk-flex-grow, 1) 1 calc(69.8\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-sqrt-3-sqrt-7\"><pre>\u7b54\u6848:\\sqrt{3}+\\sqrt{7};\u63d0\u793a:OA=2,OA_1=\\sqrt{3}.~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-bf69f8c\" data-v=\"4\" data-block-id=\"bf69f8c\"><style>@media screen and (min-width:690px){.stk-bf69f8c {flex:var(--stk-flex-grow, 1) 1 calc(30.200000000000003\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/wjq14.png\" alt=\"\" class=\"wp-image-4877\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>6.\u5982\u56fe, \u5728\u6b63\u56db\u68f1\u53f0  A B C D-A_{1} B_{1} C_{1} D_{1}  \u4e2d,  A B=4, A_{1} B_{1}=2 , \u82e5\u534a\u5f84\u4e3a  r  \u7684\u7403  O  \u4e0e\u8be5~~~\\\\\u6b63\u56db\u68f1\u53f0\u7684\u5404\u4e2a\u9762\u5747\u76f8\u5207\uff0c\u6c42\u8be5\u7403\u7684\u8868\u9762\u79ef  S.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-2e6074d\" data-block-id=\"2e6074d\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-2e6074d-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-b64d268\" data-v=\"4\" data-block-id=\"b64d268\"><style>@media screen and (min-width:690px){.stk-b64d268 {flex:var(--stk-flex-grow, 1) 1 calc(69.9\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-8a9c6b8\" data-block-id=\"8a9c6b8\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-8a9c6b8-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-cd47b51\" data-v=\"4\" data-block-id=\"cd47b51\"><style>@media screen and (min-width:690px){.stk-cd47b51 {flex:var(--stk-flex-grow, 1) 1 calc(61\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d111\"><pre>\u7b54\u6848:8\\pi;\u63d0\u793a:\u5207\u70b9\u7684\u8f74\u622a\u9762\\\\\u5982\u56fe\u6240\u793a\uff0cAB=BC=1,~~~~~\\\\ED=DC=2,\\Rightarrow BF=2\\sqrt{2}.<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-39bfaa1\" data-v=\"4\" data-block-id=\"39bfaa1\"><style>@media screen and (min-width:690px){.stk-39bfaa1 {flex:var(--stk-flex-grow, 1) 1 calc(39\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjh15-1-2.png\" alt=\"\" class=\"wp-image-4878\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-73e3d05\" data-v=\"4\" data-block-id=\"73e3d05\"><style>@media screen and (min-width:690px){.stk-73e3d05 {flex:var(--stk-flex-grow, 1) 1 calc(30.1\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"307\" height=\"232\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/pmxl1-21.png\" alt=\"\" class=\"wp-image-3862\" srcset=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/pmxl1-21.png 307w, http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/pmxl1-21-300x227.png 300w\" sizes=\"auto, (max-width: 307px) 100vw, 307px\" \/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(2)\u5df2\u77e5\u4e00\u4e2a\u5706\u53f0\u7684\u4e0a\u3001\u4e0b\u5e95\u9762\u534a\u5f84\u5206\u522b\u4e3a1\u548c4,\u9ad8\u4e3a3\\sqrt{3}.\u82e5\u8be5\u5706\u53f0\u5185\u6709\u4e00\u4e2a\u7403,\u5219~~~~\\\\\u8be5\u7403\u7684\u8868\u9762\u79ef\u7684\u6700\u5927\u503c\u4e3a(\\quad)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA. 9\\pi\\quad\nB. \\frac{64\\pi}{3}\\quad\nC. 27\\pi\\quad\nD. \\frac{27\\sqrt{3}\\pi}{2}\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<p id=\"htoc-1111111111111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-759133b\" data-block-id=\"759133b\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-759133b-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-9898569\" data-v=\"4\" data-block-id=\"9898569\"><style>@media screen and (min-width:690px){.stk-9898569 {flex:var(--stk-flex-grow, 1) 1 66.7\n<p id=\"htoc-11111111111111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-b-b-frac-pi-3-rightarrow-r-frac-4-sqrt-3-3-s-frac-64-pi-3-quad\"><pre>\u7b54\u6848:B;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u63d0\u793a:\u9996\u5148\u5224\u65ad\u5706\u53f0\u9ad8\u4e8e\u7403\u7684\u76f4\u5f84,\u6c42\u51fa\u89d2B=\\frac{\\pi}{3},~~\\\\\n\\Rightarrow r= \\frac{4\\sqrt{3}}{3},S=\\frac{64\\pi}{3}\\quad.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<p id=\"htoc-111111111111111111111111111111111111111111111111111111111111111111\"><\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-bb4f1bd\" data-v=\"4\" data-block-id=\"bb4f1bd\"><style>@media screen and (min-width:690px){.stk-bb4f1bd {flex:var(--stk-flex-grow, 1) 1 33.3\n<p id=\"htoc-1111111111111111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjh7-2-2.png\" alt=\"\" class=\"wp-image-4601\" style=\"width:169px;height:auto\"\/><\/figure>\n\n\n\n<p id=\"htoc-11111111111111111111111111111111111111111111111111111111111111111111\"><\/p>\n<\/div><\/div><\/div>\n<\/div><\/div>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>7.\u5df2\u77e5\u5706\u9525\u7684\u5e95\u9762\u534a\u5f84\u4e3a 1 , \u6bcd\u7ebf\u957f\u4e3a 3 , \u6c42\u8be5\u5706\u9525\u5185\u534a\u5f84\u6700\u5927\u7684\u7403\u7684\u4f53\u79ef\u4e3a.~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-60bcbd5\" data-block-id=\"60bcbd5\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-60bcbd5-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-36583aa\" data-v=\"4\" data-block-id=\"36583aa\"><style>@media screen and (min-width:690px){.stk-36583aa {flex:var(--stk-flex-grow, 1) 1 calc(80.2\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-da\"><pre>\u7b54\u6848:\\frac{\\sqrt{2}}{3} \\pi ;\u63d0\u793a: P D=2 \\sqrt{2}, \\triangle P E O \\backsim \\triangle P D B , \u6545  \\frac{P O}{P B}=\\frac{O E}{D B} , ~\\\\\u5373  \\frac{2 \\sqrt{2}-r}{3}=\\frac{r}{1} , \u89e3\u5f97  r=\\frac{\\sqrt{2}}{2} ,\n\nV=\\frac{\\sqrt{2}}{3} \\pi.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-c5c3166\" data-v=\"4\" data-block-id=\"c5c3166\"><style>@media screen and (min-width:690px){.stk-c5c3166 {flex:var(--stk-flex-grow, 1) 1 calc(19.8\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/02\/ltjh16.png\" alt=\"\" class=\"wp-image-4513\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-palette-color-2-color has-alpha-channel-opacity has-palette-color-2-background-color has-background is-style-wide\"\/>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-11111111111111111111111111\">\u5e73\u884c\u4e0e\u5782\u76f4<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.(1)\u5982\u56fe\u6240\u793a, \u5728\u56db\u68f1\u96c9  P-A B C D  \u4e2d, \u56db\u8fb9\u5f62  A B C D  \u662f\u5e73\u884c\u56db\u8fb9\u5f62,  M  \u662f  P C \u7684\u4e2d\u70b9,  ~~\\\\\u5728 D M  \u4e0a\u53d6\u4e00\u70b9  G , \u8fc7  G  \u548c  P A  \u4f5c\u5e73\u9762\u4ea4  B D  \u4e8e\u70b9  H . \u6c42\u8bc1:  P A \/ \/ G H .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-22b3d1a\" data-block-id=\"22b3d1a\"><style>.stk-22b3d1a {margin-bottom:13px !important;}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-22b3d1a-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-ff7312f\" data-v=\"4\" data-block-id=\"ff7312f\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-ff7312f-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-ff7312f-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-p-a-b-m-d-p-a-h-g-cap-b-m-d-g-h-p-a-subset-p-a-h-g-therefore-p-a-g-h\"><pre>\u5148\u8bc1:  P A \/ \/  \u5e73\u9762  B M D ,\\\\\u518d\u5229\u7528\u7ebf\u9762\u5e73\u884c\u7684\u6027\u8d28\\\\\u5b9a\u7406\u8bc1\u660eP A \/ \/ G H .~~~~~~<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-711d888\" data-v=\"4\" data-block-id=\"711d888\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-711d888-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-711d888-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh18-2.png\" alt=\"\" style=\"width:202px;height:auto\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-1c4295d\" data-v=\"4\" data-block-id=\"1c4295d\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-1c4295d-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-1c4295d-inner-blocks\">\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh18-1.png\" alt=\"\" style=\"width:211px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(2)\u5982\u56fe,\u5728\u56db\u68f1\u9525  S-A B C D  \u4e2d, B C \/ \/ A D,\u70b9  F  \u5728\u7ebf\u6bb5  S E  \u4e0a,\u4e14  B F \/ \/  \u5e73\u9762  S C D.~~~~~~~~~~\\\\\u6c42\u8bc1: F  \u4e3a\u7ebf\u6bb5  S E  \u7684\u4e2d\u70b9.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-4fe8c37\" data-block-id=\"4fe8c37\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-4fe8c37-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-04623a3\" data-v=\"4\" data-block-id=\"04623a3\"><style>@media screen and (min-width:690px){.stk-04623a3 {flex:var(--stk-flex-grow, 1) 1 calc(66.667\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-ti1\"><pre>\u63d0\u793a:\u6027\u8d28\u5b9a\u7406\\Rightarrow BF\/\/CG,BC\/\/FG.~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjhxz1.png\" alt=\"\" class=\"wp-image-4779\" style=\"width:235px;height:auto\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-b35e04f\" data-v=\"4\" data-block-id=\"b35e04f\"><style>@media screen and (min-width:690px){.stk-b35e04f {flex:var(--stk-flex-grow, 1) 1 calc(33.333\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhxz0.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2.\n\u5982\u56fe, \u5728\u957f\u65b9\u4f53  A B C D-A_{1} B_{1} C_{1} D_{1}  \u4e2d,  A D=1, A B=\\sqrt{3}, E, F, G  \u5206\u522b\u4e3a  A B, ~~~~~~~~~~~\\\\B C, C_{1} D_{1}  \u7684\u4e2d\u70b9, \u70b9  P  \u5728\u5e73\u9762  A B C D  \u5185, \u82e5\u76f4\u7ebf  D_{1} P \/ \/  \u5e73\u9762  E F G , \u6c42\u70b9  P \u7684\u8f68\u8ff9\u7684\u957f\u5ea6.~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-3e26234\" data-block-id=\"3e26234\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-3e26234-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-a804417\" data-v=\"4\" data-block-id=\"a804417\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-a804417-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-a804417-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d11111\"><pre>\u7b54\u6848:2;\u63d0\u793a:\u70b9P\u7684\u8f68\\\\\u8ff9\u4e3a\u7ebf\u6bb5AC,|AC|=2.<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-49f84b9\" data-v=\"4\" data-block-id=\"49f84b9\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-49f84b9-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-49f84b9-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-11111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh19-2.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-0849052\" data-v=\"4\" data-block-id=\"0849052\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-0849052-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-0849052-inner-blocks\">\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh19-1.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3. \u5982\u56fe\u6240\u793a, \u5df2\u77e5\u56db\u8fb9\u5f62  A B C D  \u662f\u6b63\u65b9\u5f62, \u56db\u8fb9\u5f62  A C E F  \u662f\u77e9\u5f62,  M  \u662f\u7ebf\u6bb5  E F  \u7684\u4e2d\u70b9.~~~~\\\\\n(1)\u6c42\u8bc1:  A M \/ \/  \u5e73\u9762  B D E ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u82e5\u5e73\u9762  A D M \\cap  \u5e73\u9762  B D E=l , \u5e73\u9762  A B M \\cap  \u5e73\u9762  B D E=m , \u8bd5\u5206\u6790  l  \u4e0e  m  \u7684\u4f4d\u7f6e\u5173~~\\\\\u7cfb, \u5e76\u8bc1\u660e\u4f60\u7684\u7ed3\u8bba.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-cf796fe\" data-block-id=\"cf796fe\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-cf796fe-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-d688880\" data-v=\"4\" data-block-id=\"d688880\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-d688880-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-d688880-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-c\"><pre>\u5145\u5206\u5229\u7528\u7ebf\u9762\u5e73\u884c\u7684\u6027\\\\\u8d28\u5b9a\u7406,\u7531 (1)\u77e5  A M \/ \/  ~~\\\\\u5e73\u9762  B D E ,\\Rightarrow  l \/ \/ A M ,~\\\\\n\u540c\u7406,    m \/ \/ A M , \\Rightarrow  l \/ \/ m .<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-ea3083c\" data-v=\"4\" data-block-id=\"ea3083c\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-ea3083c-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-ea3083c-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-1111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh15-2.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-a0c8ced\" data-v=\"4\" data-block-id=\"a0c8ced\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-a0c8ced-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-a0c8ced-inner-blocks\">\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh15-3.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4.(1)\u5982\u56fe, \u51e0\u4f55\u4f53  E-A B C D  \u662f\u56db\u68f1\u9525,  \\triangle A B D  \u4e3a\u6b63\u4e09\u89d2\u5f62,  C B=C D, E C \\perp B D .~~~~~~\\\\\n(1) \u6c42\u8bc1:  B E=D E ;\n( 2) \u82e5  \\angle B C D=120^{\\circ}, M  \u4e3a\u7ebf\u6bb5  A E  \u7684\u4e2d\u70b9,\n\u6c42\u8bc1:  D M \/ \/  \u5e73\u9762  B E C .<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-03e0200\" data-block-id=\"03e0200\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-03e0200-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-c92230b\" data-v=\"4\" data-block-id=\"c92230b\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-c92230b-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-c92230b-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-x\"><pre>(2)\u8bc1\u660e\u5e73\u9762MDN\/\/~~~\\\\\u5e73\u9762EBC.~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-35613e4\" data-v=\"4\" data-block-id=\"35613e4\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-35613e4-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-35613e4-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-111111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh19-3.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-66bfd61\" data-v=\"4\" data-block-id=\"66bfd61\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-66bfd61-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-66bfd61-inner-blocks\">\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh19.png\" alt=\"\" style=\"width:216px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(2)\u5982\u56fe,\u6b63\u65b9\u4f53\u53ca\u76f8\u5e94\u8fb9\u7684\u4e2d\u70b9,\u6c42\u8bc1:MN\/\/\u5e73\u9762ABC.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-15f8c9c\" data-block-id=\"15f8c9c\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-15f8c9c-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-a7cc270\" data-v=\"4\" data-block-id=\"a7cc270\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-a7cc270-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-a7cc270-inner-blocks\">\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhpx1.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-17320a8\" data-v=\"4\" data-block-id=\"17320a8\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-17320a8-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-17320a8-inner-blocks\">\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhpx2.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-434d327\" data-v=\"4\" data-block-id=\"434d327\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-434d327-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-434d327-inner-blocks\"><\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-ti\"><pre>\u63d0\u793a:\u8f6c\u5316\u4e3a\u9762\u9762\u5e73\u884c.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5.\u5982\u56fe,  P A  \u5782\u76f4\u4e8e\u4ee5  A B  \u4e3a\u76f4\u5f84\u7684\u5706\u6240\u5728\u5e73\u9762,  C  \u4e3a\u5706\u4e0a\u5f02\u4e8e  A, B  \u7684\u4efb\u610f\u4e00\u70b9,  A E \\perp P C , \\\\\u5782\u8db3\u4e3a  E , \u70b9  F  \u662f  P B  \u4e0a\u4e00\u70b9, \u6c42\u8bc1: (1)  B C \\perp  \u5e73\u9762  P A C ;(2)  A E \\perp E F .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-41bad4b\" data-block-id=\"41bad4b\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-41bad4b-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-a89a090\" data-v=\"4\" data-block-id=\"a89a090\"><style>@media screen and (min-width:690px){.stk-a89a090 {flex:var(--stk-flex-grow, 1) 1 calc(75\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d111111\"><pre>\u63d0\u793a:(1)  P A \\perp B C, A C \\perp B C, P A \\cap A C=A ~~~~~~~~~~~~~~~~~~~~~~~~\\\\\\Rightarrow B C \\perp  \u5e73\u9762  P A C ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2) \u7531 (1) \u77e5  B C \\perp A E, A E \\perp P C , \u4e14  B C \\cap P C=C~~~~~~~~~~~~~~~~~~~\\\\\\Rightarrow A E \\perp  \u5e73\u9762  P C B .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-07049cd\" data-v=\"4\" data-block-id=\"07049cd\"><style>@media screen and (min-width:690px){.stk-07049cd {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhcz1.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>6.\u5982\u56fe\u6240\u793a, \u5728\u56db\u68f1\u96c9  P-A B C D  \u4e2d,  P A \\perp  \u5e95\u9762  A B C D, A B \\perp A D, A C \\perp C D, \\angle A B C\\\\   =60^{\\circ}, P A=A B=B C, E  \u662f  P C  \u7684\u4e2d\u70b9.\u8bc1\u660e:\n(1)  C D \\perp A E ;\n(2)  P D \\perp  \u5e73\u9762  A B E .~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-6f1e938\" data-block-id=\"6f1e938\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-6f1e938-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-7e1ffbf\" data-v=\"4\" data-block-id=\"7e1ffbf\"><style>@media screen and (min-width:690px){.stk-7e1ffbf {flex:var(--stk-flex-grow, 1) 1 66.7\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-t\"><pre>\u63d0\u793a: (1)  P A \\perp C D, A C \\perp C D, P A \\cap A C=A ,~~~~~~~~~\\\\\n\\Rightarrow C D \\perp  \u5e73\u9762  P A C .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2) \u7531  A E \\perp  \u5e73\u9762  P C D \\Rightarrow A E \\perp P D , \u53c8  A B \\perp P D,~~~~~\\\\ A B \\cap A E=A, \\therefore P D \\perp  \u5e73\u9762  A B E .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-573536e\" data-v=\"4\" data-block-id=\"573536e\"><style>@media screen and (min-width:690px){.stk-573536e {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhcz2.png\" alt=\"\" style=\"width:202px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>7.\u5982\u56fe,\u68f1\u957f\u4e3a 2 \u7684\u6b63\u65b9\u4f53  A B C D-A_{1} B_{1} C_{1} D_{1}  \u4e2d, P, Q  \u5206\u522b\u662f\u68f1  C C_{1}  ,\u6821  B C  \u7684\u4e2d\u70b9,\u52a8~\\\\\u70b9  M  \u6ee1\u8db3  \\overrightarrow{D M}=\\lambda \\overrightarrow{D A}+\\mu \\overrightarrow{D D_{1}}  ,\u5176\u4e2d  \\lambda, \\mu \\in R  ,\u5219\u4e0b\u5217\u7ed3\u8bba\u6b63\u786e\u7684\u662f(\\quad)~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.\u82e5  \\lambda+\\mu=1  ,\u5219  C M \\perp D B_{1} \\quad~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nB.\u82e5  \\lambda=\\mu  ,\u5219\u4e09\u68f1\u96c9  B_{1}-A M C  \u7684\u4f53\u79ef\u4e3a\u5b9a\u503c\\quad~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nC.\u82e5  \\mu=\\frac{1}{2}, 0 \\leq \\lambda \\leq 1  ,\u5219\u76f4\u7ebf  P M  \u4e0e\u76f4\u7ebf  B C  \u6240\u6210\u89d2\u7684\u6700\u5c0f\u503c\u4e3a  60^{\\circ} \\quad~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nD.\u82e5\u52a8\u70b9  M  \u5728\u4e09\u68f1\u9525  C-D P Q  \u5916\u63a5\u7403\u7684\u8868\u9762\u4e0a,\u5219\u70b9  M  \u7684\u8f68\u8ff9\u957f\u5ea6\u4e3a  \\sqrt{2} \\pi ~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-11111111111111111111111111111111111111111111111111111111111111\"><pre>\u7b54\u6848:ABD;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-db36cc0\" data-block-id=\"db36cc0\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-db36cc0-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-3b1e4d2\" data-v=\"4\" data-block-id=\"3b1e4d2\"><style>@media screen and (min-width:690px){.stk-3b1e4d2 {flex:var(--stk-flex-grow, 1) 1 25\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjhzft-1.png\" alt=\"\" class=\"wp-image-4774\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-7dc1d3b\" data-v=\"4\" data-block-id=\"7dc1d3b\"><style>@media screen and (min-width:690px){.stk-7dc1d3b {flex:var(--stk-flex-grow, 1) 1 25\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjhzft-2.png\" alt=\"\" class=\"wp-image-4775\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-3e0a4ef\" data-v=\"4\" data-block-id=\"3e0a4ef\"><style>@media screen and (min-width:690px){.stk-3e0a4ef {flex:var(--stk-flex-grow, 1) 1 25\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjhzft-3.png\" alt=\"\" class=\"wp-image-4776\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-ce94847\" data-v=\"4\" data-block-id=\"ce94847\"><style>@media screen and (min-width:690px){.stk-ce94847 {flex:var(--stk-flex-grow, 1) 1 25\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjhzft-4.png\" alt=\"\" class=\"wp-image-4777\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n<\/details>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-palette-color-2-color has-alpha-channel-opacity has-palette-color-2-background-color has-background is-style-wide\"\/>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-1111111111111111111111111111111111111111\">\u4ea4\u7ebf\u53ca\u622a\u9762\u95ee\u9898<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.(1)\u5df2\u77e5\u76f4\u56db\u68f1\u67f1  A B C D-A_{1} B_{1} C_{1} D_{1}  \u7684\u68f1\u957f\u5747\u4e3a  2, \\angle B A D   =60^{\\circ},\u6c42\u4ee5  D_{1}  \u4e3a\u7403\u5fc3,  ~~~~~\\\\\\sqrt{5} \u4e3a\u534a\u5f84\u7684\u7403\u9762\u4e0e\u4fa7\u9762  B C C_{1} B_{1}  \u7684\u4ea4\u7ebf\u957f.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(2)\u5df2\u77e5\u6b63\u65b9\u4f53  A B C D-A_{1} B_{1} C_{1} D_{1}  \u7684\u68f1\u957f\u5747\u4e3a  2, \u6c42\u4ee5  D_{1}  \u4e3a\u7403\u5fc3,  \\sqrt{5} \u4e3a\u534a\u5f84\u7684\u7403\u9762\u4e0e\u5e95\\\\\u9762  ABCD  \u7684\u4ea4\u7ebf\u957f.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-button-group stk-block-button-group stk-block stk-5eeb192\" data-block-id=\"5eeb192\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-button-group\">\n<div class=\"wp-block-stackable-button stk-block-button stk-block stk-e792da1\" data-block-id=\"e792da1\"><style>.stk-e792da1 {border-top-left-radius:13px !important;border-top-right-radius:13px !important;border-bottom-right-radius:13px !important;border-bottom-left-radius:13px !important;overflow:hidden !important;}<\/style><a class=\"stk-link stk-button stk--hover-effect-darken\" href=\"https:\/\/ggb123.cn\/calculator\/z86h6s7w\" target=\"_blank\" rel=\"noreferrer noopener\"><span class=\"stk-button__inner-text\">\u753b\u677f\u5b9e\u9a8c<\/span><\/a><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-e761ef1\" data-block-id=\"e761ef1\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-e761ef1-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-fff1b87\" data-v=\"4\" data-block-id=\"fff1b87\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-fff1b87-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-fff1b87-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d1111111111\"><pre>\u7b54\u6848:(1)\\frac{\\sqrt{2}\\pi}{2},(2)\\frac{\\pi}{2}.~~~<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-6058067\" data-v=\"4\" data-block-id=\"6058067\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-6058067-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-6058067-inner-blocks\">\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx1.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-9d6b960\" data-v=\"4\" data-block-id=\"9d6b960\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-9d6b960-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-9d6b960-inner-blocks\">\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx1-1.png\" alt=\"\" style=\"width:182px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(3)\u5982\u56fe, \u5728\u957f\u65b9\u4f53  A B C D-A_{1} B_{1} C_{1} D_{1}  \u4e2d,  A D=A A_{1}=6, A B=3 \\sqrt{3}, E, F  \u5206\u522b\u5728\u68f1~~\\\\  A_{1} B_{1}, C D  \u4e0a, \u4e14  A_{1} E=C F=\\sqrt{3} , \u6c42\u4ee5  E F  \u4e3a\u76f4\u5f84\u7684\u7403\u4e0e\u4fa7\u9762  A_{1} A D D_{1}  \u7684\u4ea4\u7ebf\u957f.~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-17bcf3a\" data-block-id=\"17bcf3a\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-17bcf3a-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-3f36632\" data-v=\"4\" data-block-id=\"3f36632\"><style>@media screen and (min-width:690px){.stk-3f36632 {flex:var(--stk-flex-grow, 1) 1 66.7\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjhjx1-1-1.png\" alt=\"\" class=\"wp-image-4787\" style=\"width:365px;height:auto\"\/><\/figure>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d1111111111111\"><pre>\u7b54\u6848:\\frac{4\\sqrt{3}\\pi}{3};\u63d0\u793a:OM=\\frac{3\\sqrt{3}\\pi}{2},ON=\\frac{5\\sqrt{3}\\pi}{2}\\Rightarrow~~~~\\\\MN=2\\sqrt{3},\u4ea4\u7ebf\u7ec4\u5408\u540e\u4e3a\u5706\u5fc3\u89d2\u4e3a\\frac{2\\pi}{3}\u7684\u6247\u5f62\uff0c\u6240\u4ee5\\\\\u4ea4\u7ebf\u957f\u4e3a\\frac{4\\sqrt{3}\\pi}{3}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-cbbf8bd\" data-v=\"4\" data-block-id=\"cbbf8bd\"><style>@media screen and (min-width:690px){.stk-cbbf8bd {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjh1-1-1-2.png\" alt=\"\" style=\"width:189px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2.\u5df2\u77e5\u5728\u5706\u67f1  O_{1} O_{2}  \u5185\u6709\u4e00\u4e2a\u7403  O , \u8be5\u7403\u4e0e\u5706\u67f1\u7684\u4e0a\u3001\u4e0b\u5e95\u9762\u53ca\u6bcd\u7ebf\u5747\u76f8\u5207,\u8fc7\u76f4\u7ebf  O_{1} O_{2}  \\\\\u7684\u5e73\u9762\u622a\u5706\u67f1\u5f97\u5230\u56db\u8fb9\u5f62  A B C D , \u5176\u9762\u79ef\u4e3a 8,\u82e5  P \u4e3a\u5706\u67f1\u5e95\u9762\u5706\u5f27  \\stackrel\\frown{CD}  \u7684\u4e2d\u70b9, \u6c42\u5e73\u9762\\\\  P A B  \u4e0e\u7403  O  \u7684\u4ea4\u7ebf\u957f.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-button-group stk-block-button-group stk-block stk-6342fd1\" data-block-id=\"6342fd1\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-button-group\">\n<div class=\"wp-block-stackable-button stk-block-button stk-block stk-45581c0\" data-block-id=\"45581c0\"><style>.stk-45581c0 {border-top-left-radius:13px !important;border-top-right-radius:13px !important;border-bottom-right-radius:13px !important;border-bottom-left-radius:13px !important;overflow:hidden !important;}<\/style><a class=\"stk-link stk-button stk--hover-effect-darken\" href=\"https:\/\/ggb123.cn\/m\/varv55dg\" target=\"_blank\" rel=\"noreferrer noopener\"><span class=\"stk-button__inner-text\">\u753b\u677f\u5b9e\u9a8c<\/span><\/a><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-04f705c\" data-block-id=\"04f705c\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-04f705c-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f947784\" data-v=\"4\" data-block-id=\"f947784\"><style>@media screen and (min-width:690px){.stk-f947784 {flex:var(--stk-flex-grow, 1) 1 calc(47.3\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848:<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-bigtriangleup-oo-1n-sim-bigtriangleup-amn-rightarrow-o-1n-r-frac-2-sqrt-10-5-frac-2-sqrt-10-pi-5\"><pre>\u7b54\u6848:\\frac{4\\sqrt{10} \\pi }{5};~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u63d0\u793a:\u7531\\bigtriangleup OO_1N\\sim \\bigtriangleup AMN,~~~~~~~~~\\\\\\Rightarrow O_1N=r=\\frac{2\\sqrt{10} }{5} ,\u4ea4\u7ebf\u957f\\frac{4\\sqrt{10} \\pi }{5}.<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-7b799d4\" data-v=\"4\" data-block-id=\"7b799d4\"><style>@media screen and (min-width:690px){.stk-7b799d4 {flex:var(--stk-flex-grow, 1) 1 calc(27.69999999999999\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx4-1.png\" alt=\"\" style=\"width:121px;height:auto\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-5b3eb1f\" data-v=\"4\" data-block-id=\"5b3eb1f\"><style>@media screen and (min-width:690px){.stk-5b3eb1f {flex:var(--stk-flex-grow, 1) 1 calc(25.000000000000014\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx4.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.\u5982\u56fe, \u5728\u6b63\u65b9\u4f53  A B C D-A_{1} B_{1} C_{1} D_{1}  \u4e2d, \u70b9  E, F  \u5206\u522b\u662f\u68f1  B_{1} B, B_{1} C_{1}  \u7684\u4e2d\u70b9,\u70b9  G  \u662f\u68f1~~\\\\  C_{1} C  \u7684\u4e2d\u70b9, \u5219\u8fc7\u7ebf\u6bb5  A G  \u4e14\u5e73\u884c\u4e8e\u5e73\u9762  A_{1} E F  \u7684\u622a\u9762\u56fe\u5f62\u4e3a(\\quad )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA. \u77e9\u5f62\\quad \nB.\u4e09\u89d2\u5f62\\quad \nC.\u6b63\u65b9\u5f62\\quad \nD. \u7b49\u8170\u68af\u5f62~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-c7ecbc6\" data-block-id=\"c7ecbc6\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-c7ecbc6-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-14f2e93\" data-v=\"4\" data-block-id=\"14f2e93\"><style>@media screen and (min-width:690px){.stk-14f2e93 {flex:var(--stk-flex-grow, 1) 1 calc(71\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-2cc8dda\" data-block-id=\"2cc8dda\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-2cc8dda-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-4e45a2f\" data-v=\"4\" data-block-id=\"4e45a2f\"><style>@media screen and (min-width:690px){.stk-4e45a2f {flex:var(--stk-flex-grow, 1) 1 calc(55.19999999999999\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d1111111\"><pre>\u7b54\u6848:D.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-15f932c\" data-v=\"4\" data-block-id=\"15f932c\"><style>@media screen and (min-width:690px){.stk-15f932c {flex:var(--stk-flex-grow, 1) 1 calc(44.80000000000001\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjhjx3-1.png\" alt=\"\" class=\"wp-image-3929\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-08b3da3\" data-v=\"4\" data-block-id=\"08b3da3\"><style>@media screen and (min-width:690px){.stk-08b3da3 {flex:var(--stk-flex-grow, 1) 1 calc(29\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx3-2.png\" alt=\"\" style=\"width:192px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4. \u5df2\u77e5\u6b63\u65b9\u4f53  A B C D-A_{1} B_{1} C_{1} D_{1}  \u7684\u68f1\u957f\u4e3a  a , \u70b9  E, F, G  \u5206\u522b\u4e3a\u68f1  A B, A A_{1}, C_{1} D_{1} \u7684~~~~~\\\\\u4e2d\u70b9,\u8fc7  E, F, G  \u4e09\u70b9\u4f5c\u6b63\u65b9\u4f53\u7684\u622a\u9762, \u6c42\u6b64\u622a\u9762\u7684\u9762\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-ec90303\" data-block-id=\"ec90303\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-ec90303-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f439d84\" data-v=\"4\" data-block-id=\"f439d84\"><style>@media screen and (min-width:690px){.stk-f439d84 {flex:var(--stk-flex-grow, 1) 1 calc(43.599999999999994\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d11111111\"><pre>\u7b54\u6848:\\frac{3\\sqrt{3} }{4} a^2;\u56fe\u4e2d\u6b63\u516d\u8fb9\u5f62\u7684~~\\\\\u9762\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-1e15972\" data-v=\"4\" data-block-id=\"1e15972\"><style>@media screen and (min-width:690px){.stk-1e15972 {flex:var(--stk-flex-grow, 1) 1 calc(28.3\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx2-2.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-e1edef0\" data-v=\"4\" data-block-id=\"e1edef0\"><style>@media screen and (min-width:690px){.stk-e1edef0 {flex:var(--stk-flex-grow, 1) 1 calc(28.1\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx2-1.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u7ec3\u4e60.\u5df2\u77e5\u6b63\u65b9\u4f53  A B C D-A_{1} B_{1} C_{1} D_{1}  \u7684\u68f1\u957f\u4e3a  4, E, F  \u5206\u522b\u662f\u68f1  A A_{1} ,  B C  \u7684\u4e2d\u70b9, \u753b\u51fa~~\\\\ \u5e73\u9762 D_{1} E F  \u622a\u8be5\u6b63\u65b9\u4f53\u6240\u5f97\u7684\u622a\u9762.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-2d101cf\" data-block-id=\"2d101cf\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-2d101cf-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-9e273df\" data-v=\"4\" data-block-id=\"9e273df\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-9e273df-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-9e273df-inner-blocks\"><\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-456f4be\" data-v=\"4\" data-block-id=\"456f4be\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-456f4be-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-456f4be-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx6-2.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-179d265\" data-v=\"4\" data-block-id=\"179d265\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-179d265-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-179d265-inner-blocks\">\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx6-1.png\" alt=\"\" style=\"width:208px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5.\u5982\u56fe, \u6b63\u65b9\u4f53  A B C D-A_{1} B_{1} C_{1} D_{1}  \u7684\u68f1\u957f\u4e3a  2, E, F  \u5206\u522b\u4e3a  A A_{1}, A B  \u7684\u4e2d\u70b9,  M  \u70b9\u662f\u6b63~~\\\\\u65b9\u5f62  A B B_{1} A_{1}  \u5185\u7684\u52a8\u70b9, \u82e5  C_{1} M \/ \/  \u5e73\u9762  C D_{1} E F , \u6c42  M  \u70b9\u7684\u8f68\u8ff9\u957f\u5ea6.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-2f52581\" data-block-id=\"2f52581\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-2f52581-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f6b909d\" data-v=\"4\" data-block-id=\"f6b909d\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-f6b909d-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-f6b909d-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d11111111111\"><pre>\u7b54\u6848:\\sqrt{2}.~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-4f0ede5\" data-v=\"4\" data-block-id=\"4f0ede5\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-4f0ede5-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-4f0ede5-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx7-2.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-395229a\" data-v=\"4\" data-block-id=\"395229a\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-395229a-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-395229a-inner-blocks\">\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx7-1.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u7ec3\u4e60. \u5982\u56fe, \u5728\u68f1\u957f\u4e3a 1 \u7684\u6b63\u65b9\u4f53  A B C D-A_{1} B_{1} C_{1} D_{1}  \u4e2d,  M, N  \u5206\u522b\u662f  A_{1} D_{1}, A_{1} B_{1}  \u7684\u4e2d\u70b9, \\\\\u8fc7\u76f4\u7ebf  B D  \u7684\u5e73\u9762  \\alpha \/ \/  \u5e73\u9762  A M N , \u6c42\u5e73\u9762  \\alpha  \u622a\u8be5\u6b63\u65b9\u4f53\u6240\u5f97\u622a\u9762\u7684\u9762\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-fc45c0c\" data-block-id=\"fc45c0c\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-fc45c0c-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-fec6b87\" data-v=\"4\" data-block-id=\"fec6b87\"><style>@media screen and (min-width:690px){.stk-fec6b87 {flex:var(--stk-flex-grow, 1) 1 calc(36.1\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d111111111\"><pre>\u7b54\u6848:\\frac{9}{8}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-4b5f7dc\" data-v=\"4\" data-block-id=\"4b5f7dc\"><style>@media screen and (min-width:690px){.stk-4b5f7dc {flex:var(--stk-flex-grow, 1) 1 calc(33.699999999999974\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx5-2.png\" alt=\"\" style=\"width:184px;height:auto\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-c45330f\" data-v=\"4\" data-block-id=\"c45330f\"><style>@media screen and (min-width:690px){.stk-c45330f {flex:var(--stk-flex-grow, 1) 1 calc(30.200000000000014\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx5-1.png\" alt=\"\" style=\"width:206px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>6.\u5df2\u77e5\u5706\u53f0  O_{1} O_{2} , \u4e0a\u4e0b\u5e95\u9762\u534a\u5f84\u5206\u522b\u4e3a  1, ~ 4  ,\u534a\u5f84\u4e3a  R  \u7684\u7403  O  \u5185\u5207\u4e8e\u5706\u53f0,\u5219(\\quad)~~~~~~~~~~~~~~\\\\\n\nA. R=2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nB.\u5706\u53f0\u4fa7\u9762\u5c55\u5f00\u56fe\u6247\u73af\u7684\u5706\u5fc3\u89d2\u4e3a  \\frac{6 \\pi}{5} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nC.\u8fc7  O_{1}  \u7684\u622a\u9762\u4e0e\u5e95\u9762\u6240\u6210\u89d2\u4e3a  60^{\\circ}  \u65f6, O_{2}  \u5230\u622a\u9762\u8ddd\u79bb\u4e3a  \\frac{\\sqrt{6}}{2} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nD.\u5728\u5706\u53f0\u5185\u653e\u4e00\u6b63\u65b9\u4f53,\u6b63\u65b9\u4f53\u53ef\u7ed5\u5176\u4e2d\u5fc3\u81ea\u7531\u8f6c\u52a8,\u5219\u8be5\u6b63\u65b9\u4f53\u68f1\u957f\u7684\u6700\u5927\u503c\u4e3a  \\frac{4 \\sqrt{3}}{3} <\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-da1\"><pre>\u7b54\u6848:ABD;\u63d0\u793a:\u67e5\u770b\u53c2\u8003\u56fe\u8c61\uff0c\u9009\u9879D\u4e2d,\u6b63\u65b9\u4f53\u7ed5\u5176\u4e2d\u5fc3\u81ea\u7531\u8f6c\u52a8\u540e\u4e3a\u7403\u4f53.~~~~~~~~<\/pre><\/div>\n<\/details>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u56fe\u8c61<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-0ef9bdd\" data-block-id=\"0ef9bdd\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-0ef9bdd-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-7c0746c\" data-v=\"4\" data-block-id=\"7c0746c\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-7c0746c-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-7c0746c-inner-blocks\">\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjxwt2.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-1995028\" data-v=\"4\" data-block-id=\"1995028\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-1995028-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-1995028-inner-blocks\">\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjxwt1.png\" alt=\"\" style=\"width:185px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-c1c4a6d\" data-v=\"4\" data-block-id=\"c1c4a6d\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-c1c4a6d-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-c1c4a6d-inner-blocks\"><\/div><\/div><\/div>\n<\/div><\/div>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>7.\u5df2\u77e5\u5706\u53f0\u7684\u9ad8\u4e3a 2 ,\u5176\u6bcd\u7ebf\u4e0e\u5e95\u9762\u6240\u6210\u7684\u89d2\u4e3a  \\frac{\\pi}{3}  ,\u4e0b\u5e95\u9762\u534a\u5f84\u662f\u4e0a\u5e95\u9762\u534a\u5f84\u7684 2 \u500d,\u5219~~\\\\\u4e0b\u5217\u7ed3\u8bba\u6b63\u786e\u7684\u662f(\\quad)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.\u8be5\u5706\u53f0\u7684\u4e0a\u5e95\u9762\u534a\u5f84\u4e3a 2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nB.\u8be5\u5706\u53f0\u7684\u4f53\u79ef\u4e3a  \\frac{56 \\pi}{9} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nC.\u8be5\u5706\u53f0\u5916\u63a5\u7403(\u5706\u53f0\u7684\u4e0a,\u4e0b\u5e95\u9762\u7684\u5706\u5468\u5747\u5728\u7403\u9762\u4e0a)\u7684\u8868\u9762\u79ef\u4e3a  \\frac{64 \\pi}{3}~~~~~~~~~~~~~~~~~~~~~~~~~ \\\\\nD.\u7528\u5e73\u9762\u622a\u8be5\u5706\u53f0,\u82e5\u6240\u622a\u56fe\u5f62\u4e3a\u692d\u5706,\u5219\u692d\u5706\u79bb\u5fc3\u7387\u53d6\u503c\u8303\u56f4\u4e3a  \\left(0, \\frac{\\sqrt{3}}{3}\\right]~~~~~~~~~~~~~~~~~ <\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" 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