{"id":1077,"date":"2022-09-15T11:06:29","date_gmt":"2022-09-15T11:06:29","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=1077"},"modified":"2023-04-28T15:29:13","modified_gmt":"2023-04-28T07:29:13","slug":"gaozhongjingsaizhongdeyixiejihedingli","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=1077","title":{"rendered":"\u9ad8\u4e2d\u7ade\u8d5b\u4e2d\u7684\u4e00\u4e9b\u51e0\u4f55\u5b9a\u7406"},"content":{"rendered":"

1. \u5f20\u89d2\u5b9a\u7406<\/a><\/span>\n\n\u5f20\u89d2\u5b9a\u7406\u662f\u5229\u7528\u6b63\u5f26\u8ba1\u7b97\u4e09\u89d2\u5f62\u9762\u79ef\u516c\u5f0f\u7684\u76f4\u63a5\u63a8\u8bba.<\/p>\n

<\/span>

\"\u5f20\u89d2\u5b9a\u7406.png\"
\u56fe 1. \u5f20\u89d2\u5b9a\u7406<\/figcaption><\/figure>
\n
\u5b9a\u7406 1<\/span> (\u5f20\u89d2\u5b9a\u7406<\/span>).<\/span> \u5982\u56fe 1<\/a>, \u5047\u8bbe\u4e09\u89d2\u5f62$ABC$\u4e2d, $BC$\u8fb9\u4e0a\u6709\u4e00\u70b9$D$, \u5219
\n\\[
\n \\frac{\\sin\\angle BAD}{AC}+\\frac{\\sin\\angle CAD}{AB}=\\frac{\\sin\\angle BAC}{ AD}.
\n\\]
\n<\/div>
\n
\n
\u8bc1\u660e<\/span>.<\/span> \u5229\u7528\u4e09\u89d2\u5f62\u7684\u9762\u79ef\u516c\u5f0f$S_{\\triangle ABC}=AB\\cdot AC\\sin \\angle BAC$, \u4ee5\u53ca$S_{\\triangle ABC}=S_{\\triangle BAD}+S_{\\triangle DAC}$\u5373\u53ef.
\n<\/div>
\n2. \u4f59\u5f26\u5b9a\u7406<\/a><\/span>\n\n\u6bd4\u8f83\u6709\u610f\u601d\u7684\u662f, \u4e09\u89d2\u5f62\u7684\u4f59\u5f26\u5b9a\u7406\u53ef\u4ee5\u76f4\u63a5\u7531\u52fe\u80a1\u5b9a\u7406\u5f97\u5230[1].
\n
\u5b9a\u7406 2<\/span> (\u4f59\u5f26\u5b9a\u7406<\/span>).<\/span> \u4e09\u89d2\u5f62$ABC$\u4e2d\u89d2$A,B,C$\u4e0e\u6240\u5bf9\u7684\u8fb9\u957f$a,b,c$\u4e4b\u95f4\u6210\u7acb
\n \\[
\n c^2=a^2+b^2-2ab\\cos\\angle C.
\n \\]
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> <\/span>
\"\u4f59\u5f26\u5b9a\u7406.png\"
\u56fe 2. \u52fe\u80a1\u5b9a\u7406\u63a8\u4f59\u5f26\u5b9a\u7406<\/figcaption><\/figure>
\n \u5982\u56fe 2<\/a>\u6240\u793a, \u5229\u7528\u52fe\u80a1\u5b9a\u7406\u5373\u53ef.
\n<\/div>
\n3. \u5706\u7684\u5185\u63a5\u56db\u8fb9\u5f62\u4f59\u5f26\u5b9a\u7406<\/a><\/span>\n\n\u5173\u4e8e\u4e09\u89d2\u5f62\u7684\u4f59\u5f26\u5b9a\u7406\u53ef\u4ee5\u63a8\u5e7f\u5230\u5185\u63a5\u56db\u8fb9\u5f62[1].
\n
\u5b9a\u7406 3<\/span> (\u5706\u7684\u5185\u63a5\u56db\u8fb9\u5f62\u4f59\u5f26\u5b9a\u7406<\/span>).<\/span> \u5706\u7684\u5185\u63a5\u56db\u8fb9\u5f62$ABCD$\u4e2d\u89d2$A,B,C,D$\u4e0e\u8fb9\u957f$a=AB,b=BC,c=CD, D=DA$\u4e4b\u95f4\u6210\u7acb
\n \\[
\n a^2+b^2=c^2+d^2-2(ab+cd)\\cos\\angle D.
\n \\]
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> <\/span>
\"\u4f59\u5f26\u5b9a\u7406\u7684\u63a8\u5e7f.png\"
\u56fe 3. \u4f59\u5f26\u5b9a\u7406\u7684\u63a8\u5e7f<\/figcaption><\/figure>
\n \u5982\u56fe 3<\/a>\u6240\u793a, \u6ce8\u610f\u5230$\\angle B+\\angle D=\\pi$, \u5229\u7528\u4e09\u89d2\u5f62$ABC$\u3001$ADC$\u7684\u4f59\u5f26\u5b9a\u7406\u5373\u53ef.
\n<\/div>
\n4. \u6258\u52d2\u5bc6\u5b9a\u7406<\/a><\/span>\n\n\u5185\u63a5\u4e8e\u5706\u7684\u56db\u8fb9\u5f62\u7684\u5404\u8fb9\u4e0e\u5bf9\u89d2\u7ebf\u4e4b\u95f4\u6709\u4e0b\u5217\u5173\u7cfb[1].
\n
\u5b9a\u7406 4<\/span> (\u6258\u52d2\u5bc6\u5b9a\u7406<\/span>).<\/span> \u5047\u8bbe$ABCD$\u4e3a\u5706\u7684\u5185\u63a5\u51f8\u56db\u8fb9\u5f62, \u5219\u4e24\u7ec4\u5bf9\u8fb9\u4e58\u79ef\u4e4b\u548c\u7b49\u4e8e\u5bf9\u89d2\u7ebf\u7684\u4e58\u79ef.
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> <\/span>
\"\u6258\u52d2\u5bc6\u5b9a\u7406.png\"
\u56fe 4. \u6258\u52d2\u5bc6\u5b9a\u7406\u7684\u8bc1\u660e<\/figcaption><\/figure>
\n \u5982\u56fe 4<\/a>\u6240\u793a, \u6ce8\u610f\u5230$\\angle B+\\angle D=\\pi$, \u5229\u7528\u4e09\u89d2\u5f62$ACD$\u7684\u4f59\u5f26\u5b9a\u7406\u5e76\u7ed3\u5408\u5706\u7684\u5185\u79ef\u56db\u8fb9\u5f62\u7684\u4f59\u5f26\u5b9a\u7406, \u5f97\u5230
\n \\begin{align*}
\n f^2&=c^2+d^2-2cd\\cos D\\\\
\n &=c^2+d^2-2cd\\frac{c^2+d^2-a^2-b^2}{2(ab+cd)}\\\\
\n &=\\frac{(ad+bc)(ac+bd)}{ab+cd},
\n \\end{align*}
\n \u540c\u7406, \u6709
\n \\[
\n e^2=\\frac{(ab+cd)(ac+bd)}{ad+bc},
\n \\]
\n\u7531\u6b64\u6613\u5f97\u7ed3\u8bba.
\n<\/div>
\n
\u6ce8\u8bb0 1<\/span>.<\/span> \u7531\u6258\u52d2\u5bc6\u5b9a\u7406\u4e5f\u53ef\u8bc1\u660e\u4e09\u89d2\u5f62\u7684\u4f59\u5f26\u5b9a\u7406, \u53c2\u8003[1].
\n<\/div>
\n5. \u4e09\u5f26\u5b9a\u7406<\/a><\/span>\n\n\u5bf9\u5706\u7684\u5185\u63a5\u56db\u8fb9\u5f62, \u6211\u4eec\u8fd8\u6709\u7c7b\u4f3c\u5f20\u89d2\u5b9a\u7406\u7684\u4e00\u4e2a\u5b9a\u7406.
\n<\/span>
\"\u4e09\u5f26\u5b9a\u7406.png\"
\u56fe 5. \u4e09\u5f26\u5b9a\u7406<\/figcaption><\/figure>
\n
\u5b9a\u7406 5<\/span> (\u4e09\u5f26\u5b9a\u7406<\/span>).<\/span> \u5982\u56fe 5<\/a>\u6240\u793a, \u5047\u8bbe\u56db\u8fb9\u5f62$ABCD$\u5185\u63a5\u4e8e\u5706, \u5219
\n \\[
\n AB\\sin\\angle CAD+AC\\sin\\angle BAD=AD\\sin\\angle BAC.
\n \\]
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> \u4ee4$a=AD$, $b=AC$, $c=AB$, $d=BD$, $e=CD$, $f=BC$. \u53c8\u4ee4$\\angle BAD=\\beta$, $\\angle CAD=\\gamma$\u5219\u53ea\u9700\u8bc1\u660e
\n \\[
\n a\\sin(\\beta+\\gamma)=b\\sin\\beta+c\\sin\\gamma.
\n \\]
\n \u5229\u7528\u6b63\u5f26\u5b9a\u7406\u5bb9\u6613\u5f97\u5230
\n \\[
\n 2R\\sin(\\beta+\\gamma)=f,\\quad
\n 2R\\sin\\beta=d,\\quad
\n 2R\\sin\\gamma=e.
\n \\]
\n \u6545\u53ea\u9700\u8bc1\u660e
\n \\[
\n af=bd+ec,
\n \\]
\n \u7531\u6258\u52d2\u5bc6\u5b9a\u7406\u77e5\u4e0a\u5f0f\u6210\u7acb.
\n<\/div>
\n6. \u5171\u8fb9\u5b9a\u7406<\/a><\/span>\n\n\u5171\u8fb9\u5b9a\u7406\u975e\u5e38\u7b80\u6d01, \u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528. \u5728\u5f20\u666f\u4e2d\u9662\u58eb\u7684\u300a\u65b0\u6982\u5ff5\u51e0\u4f55\u300b\u4e2d\u7528\u6765\u8bc1\u660e\u585e\u74e6\u5b9a\u7406\u3001\u6885\u6d85\u52b3\u65af\u5b9a\u7406\u90fd\u53ea\u7528\u4e00\u884c.
\n
\u5b9a\u7406 6<\/span> (\u5171\u8fb9\u5b9a\u7406<\/span>).<\/span> \u5047\u8bbe\u4e09\u89d2\u5f62$ABP$, $ABQ$\u5177\u6709\u516c\u5171\u7684\u5e95\u8fb9$AB$, \u82e5$PQ$\u7684\u8fde\u7ebf(\u7684\u5ef6\u957f\u7ebf)\u4ea4$AB$(\u7684\u5ef6\u957f\u7ebf)\u4e8e\u70b9$M$, \u5219
\n \\[
\n \\frac{PM}{QM}=\\frac{S_{\\triangle PAB}}{S_{\\triangle QAB}}.
\n \\]
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> \u5982\u56fe 6<\/a>\u6240\u793a, \u8bc1\u660e\u5206\u4e3a\u56db\u79cd\u60c5\u51b5, \u90fd\u662f\u663e\u7136\u7684. \u4e8b\u5b9e\u4e0a\u5229\u7528\u4e09\u89d2\u5f62\u7684\u9762\u79ef\u8ba1\u7b97\u516c\u5f0f\u4ee5\u53ca\u76f8\u4f3c\u4e09\u89d2\u5f62\u5bb9\u6613\u8bc1\u660e.
\n<\/span>
\"\u5171\u8fb9\u5b9a\u7406.png\"
\u56fe 6. \u5171\u8fb9\u5b9a\u7406<\/figcaption><\/figure>
\n<\/div>
\n7. \u585e\u74e6\u5b9a\u7406<\/a><\/span>\n\n\u6211\u4eec\u79f0\u4e09\u89d2\u5f62\u9876\u70b9\u4e0e\u5176\u5bf9\u8fb9\u6216\u5bf9\u8fb9\u5ef6\u957f\u7ebf\u4e0a\u4e00\u70b9\u7684\u76f8\u8fde\u800c\u6210\u7684\u7ebf\u6bb5\u4e3a\u585e\u74e6\u7ebf\u6bb5<\/span>.
\n<\/span>
\"\u585e\u74e6\u5b9a\u7406.png\"
\u56fe 7. \u585e\u74e6\u5b9a\u7406<\/figcaption><\/figure>
\n
\u5b9a\u7406 7<\/span> (\u585e\u74e6\u5b9a\u7406<\/span>).<\/span> \u4e09\u89d2\u5f62$ABC$\u7684\u585e\u74e6\u7ebf\u6bb5$AF$\u3001$BD$\u3001$CE$, \u5b83\u4eec\u901a\u8fc7\u540c\u4e00\u70b9$O$\u7684\u5145\u8981\u6761\u4ef6\u662f
\n \\[
\n \\frac{AE}{EB}\\frac{BF}{FC}\\frac{CD}{DA}=1.
\n \\]
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> \u5982\u56fe 7<\/a>\u6240\u793a, \u5229\u7528\u5171\u8fb9\u5b9a\u7406\u6211\u4eec\u5f97\u5230
\n \\[
\n \\frac{AE}{EB}\\frac{BF}{FC}\\frac{CD}{DA}
\n =\\frac{S_{\\triangle OCA}}{S_{\\triangle OCB}}
\n \\frac{S_{\\triangle OAB}}{S_{\\triangle OAC}}
\n \\frac{S_{\\triangle OBC}}{S_{\\triangle OBA}}
\n =1.
\n \\]
\n<\/div>
\n
\u63a8\u8bba 8<\/span> (\u89d2\u5143\u585e\u74e6\u5b9a\u7406<\/span>).<\/span> \u5047\u8bbe\u5982\u585e\u74e6\u5b9a\u7406\u4e00\u6837. \u5219
\n \\[
\n \\frac{\\sin\\angle OAC}{\\sin\\angle OBC}
\n \\frac{\\sin\\angle OBA}{\\sin\\angle OCA}
\n \\frac{\\sin\\angle OCB}{\\sin\\angle OAB}=1.
\n \\]
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> \u53ea\u9700\u6ce8\u610f\u5230
\n\\begin{align*}
\n \\frac{S_{\\triangle OCA}}{S_{\\triangle OCB}}&=\\frac{OA\\cdot AC\\sin\\angle OAC}{OB\\cdot BC\\sin\\angle OBC},\\\\
\n \\frac{S_{\\triangle OAB}}{S_{\\triangle OAC}}&=\\frac{OB\\cdot BA\\sin\\angle OBA}{OC\\cdot CA\\sin\\angle OCA},\\\\
\n \\frac{S_{\\triangle OBC}}{S_{\\triangle OBA}}&=\\frac{OC\\cdot CB\\sin\\angle OCB}{OA\\cdot AB\\sin\\angle OAB},
\n\\end{align*}
\n\u4e09\u5f0f\u76f8\u4e58\u5f97\u5230
\n\\[
\n 1=\\frac{S_{\\triangle OCA}}{S_{\\triangle OCB}}
\n \\frac{S_{\\triangle OAB}}{S_{\\triangle OAC}}
\n \\frac{S_{\\triangle OBC}}{S_{\\triangle OBA}}=
\n \\frac{\\sin\\angle OAC}{\\sin\\angle OBC}
\n \\frac{\\sin\\angle OBA}{\\sin\\angle OCA}
\n \\frac{\\sin\\angle OCB}{\\sin\\angle OAB}.
\n \\]
\n<\/div>
\n8. \u6885\u6d85\u52b3\u65af\u5b9a\u7406<\/a><\/span>\n\n\u4ece\u5c04\u5f71\u51e0\u4f55\u7684\u89c2\u70b9\u6765\u770b, \u6885\u6d85\u52b3\u65af\u5b9a\u7406\u662f\u585e\u74e6\u5b9a\u7406\u7684\u5bf9\u5076\u5b9a\u7406.
\n
\u5b9a\u7406 9<\/span> (\u6885\u6d85\u52b3\u65af\u5b9a\u7406<\/span>).<\/span> \u4e09\u89d2\u5f62$ABC$\u4e2d, \u5047\u8bbe\u70b9$L, M, N$\u5728\u8fb9$BC$\u3001$CA$\u3001$AB$\u6216\u5176\u5ef6\u957f\u7ebf\u4e0a, \u5219$L, M, N$\u5171\u7ebf\u7684\u5145\u8981\u6761\u4ef6\u662f
\n \\[
\n \\frac{AN}{NB}\\frac{BL}{LC}\\frac{CM}{MA}=1.
\n \\]
\n<\/div>
\n<\/span>
\"\u6885\u6d85\u52b3\u65af\u5b9a\u7406.png\"
\u56fe 8. \u6885\u6d85\u52b3\u65af\u5b9a\u7406<\/figcaption><\/figure>
\n
\u8bc1\u660e<\/span>.<\/span> \u5982\u56fe 8<\/a>\u6240\u793a, \u6211\u4eec\u5229\u7528\u5171\u8fb9\u5b9a\u7406\u5f97\u5230
\n \\[
\n \\frac{AN}{NB}\\frac{BL}{LC}\\frac{CM}{MA}
\n =\\frac{S_{\\triangle LMA}}{S_{\\triangle LMB}}
\n \\frac{S_{\\triangle LMB}}{S_{\\triangle LMC}}
\n \\frac{S_{\\triangle LMC}}{S_{\\triangle LMA}}
\n =1.
\n \\]
\n<\/div><\/p>\n

9. \u65af\u56fe\u5c14\u7279\u5b9a\u7406<\/a><\/span>\n\n<\/span>

\"\u65af\u56fe\u5c14\u7279\u5b9a\u7406.png\"
\u56fe 9. \u65af\u56fe\u5c14\u7279\u5b9a\u7406<\/figcaption><\/figure>
\n
\u5b9a\u7406 10<\/span> (\u65af\u56fe\u5c14\u7279\u5b9a\u7406\/\u963f\u6ce2\u7f57\u5c3c\u5965\u65af\u5b9a\u7406<\/span>).<\/span> \u5047\u8bbe$D$\u4e3a\u4e09\u89d2\u5f62$ABC$\u4e2d$BC$\u8fb9\u4e0a\u7684\u4e00\u70b9, \u89d2$A,B,C$\u5bf9\u5e94\u7684\u8fb9\u957f\u5206\u522b\u4e3a$a,b,c$, \u8fc7\u9876\u70b9$A$\u7684\u585e\u74e6\u7ebf\u6bb5$AD$\u957f\u5ea6\u4e3a$d$. \u5219
\n\\[
\n DC c^2+DB b^2=a(d^2+DB\\cdot DC).
\n\\]
\n\u7279\u522b\u5730, \u5f53$D$\u4e3a\u4e2d\u7ebf\u65f6, \u5f97\u5230\u963f\u6ce2\u7f57\u5c3c\u5965\u65af\u5b9a\u7406:
\n\\[
\n c^2+b^2=2(d^2+BD^2).
\n\\]
\n<\/div><\/p>\n

10. \u897f\u59c6\u677e\u5b9a\u7406<\/a><\/span>\n\n11. \u4e5d\u70b9\u5706\u5b9a\u7406<\/a><\/span>\n<\/p>\n

[1] \u8521\u8070\u660e\u3002\u4e94\u5408\u4e00\u5b9a\u7406\u3002\u6578\u5b78\u50b3\u64ad\u5b63\u520a, 41(4), 60-68, 2017\u3002 Available from : http:\/\/web.math.sinica.edu.tw\/math_media\/d414\/41406.pdf.<\/p>\n","protected":false},"excerpt":{"rendered":"

1. \u5f20\u89d2\u5b9a\u7406 \u5f20\u89d2\u5b9a\u7406\u662f\u5229\u7528\u6b63\u5f26\u8ba1\u7b97\u4e09\u89d2\u5f62\u9762\u79ef\u516c\u5f0f\u7684\u76f4\u63a5\u63a8\u8bba. \u56fe 1. \u5f20\u89d2\u5b9a\u7406 \u5b9a\u7406 1 … \u7ee7\u7eed\u9605\u8bfb\u9ad8\u4e2d\u7ade\u8d5b\u4e2d\u7684\u4e00\u4e9b\u51e0\u4f55\u5b9a\u7406<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[194],"class_list":["post-1077","post","type-post","status-publish","format-standard","hentry","category-math","tag-shuxue-jingsai-gaozhong-jihedingli","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1077","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1077"}],"version-history":[{"count":36,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1077\/revisions"}],"predecessor-version":[{"id":1162,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1077\/revisions\/1162"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1077"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1077"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1077"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}