{"id":290,"date":"2021-02-24T12:00:00","date_gmt":"2021-02-24T04:00:00","guid":{"rendered":"https:\/\/blog.cruciferslab.net\/?p=290"},"modified":"2024-03-08T10:48:39","modified_gmt":"2024-03-08T02:48:39","slug":"%e5%86%8d%e6%8e%a2%e9%be%8d%e5%8d%9a%e5%a3%ab%e5%9c%93%e7%9b%a4%e6%8b%bc%e7%89%87-3","status":"publish","type":"post","link":"https:\/\/blog.cruciferslab.net\/?p=290","title":{"rendered":"\u518d\u63a2\u9f8d\u535a\u58eb\u5713\u76e4\u62fc\u7247 (3)"},"content":{"rendered":"\n<p>\u7cfb\u5217\u6587\u7684\u7b2c\u4e09\u7bc7\uff0c\u4f86\u6311\u6230\u5927\u9b54\u738b 11-12-13 \u7d44\u5408\u4e86\u3002\u524d\u9762\u5169\u7bc7\u6587\u7ae0\u53ef\u4ee5\u7531<a href=\"https:\/\/blog.cruciferslab.net\/?tag=lonpos\" data-type=\"post_tag\" data-id=\"6\">\u6a19\u7c64<\/a>\u9023\u7d50\u3002<\/p>\n\n\n\n<!--more-->\n\n\n\n<h2 class=\"wp-block-heading\">\u5f9e\u7b2c\u4e94\u7a2e\u56f0\u96e3\u5ea6\u7684\u4e00\u500b\u7b54\u6848\u958b\u59cb<\/h2>\n\n\n\n<p>\u4e4b\u6240\u4ee5\u6703\u8aaa\u662f\u5927\u9b54\u738b\u662f\u56e0\u70ba\uff0c\u5b83\u5011\u7684\u9577\u5ea6\u7d44\u5408\u5176\u5be6\u662f\u500b\u5f88\u7279\u5b9a\u7684\u6578\u5b57\uff0c\u662f\u5efa\u7acb\u5728\u5176\u4ed6\u62fc\u7247\u7684\u7d44\u5408\u4e0a\u5207\u51fa\u4f86\u7684\u3002\u7576\u5e74\u6211\u7528\u7d19\u7247\u6e4a\u51fa\u4e86\u4e00\u500b\u7b2c\u4e94\u7a2e\u56f0\u96e3\u5ea6\u7684\u7b54\u6848\uff0c\u5f9e\u90a3\u88e1\u624d\u77e5\u9053\u9019\u4e09\u7247\u62fc\u7247\u7684\u5be6\u969b\u5927\u5c0f\u3002\u4ee5\u4e0b\u5c31\u662f\u9019\u7d44\u7b54\u6848\uff0c\u540c\u6642\u5c07\u4e00\u4e9b\u4ea4\u9ede\u6a19\u4e0a\u5b57\u6bcd\uff0c\u4e0b\u9762\u6709\u7528\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"600\" height=\"611\" src=\"https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/01\/Sol74b.png\" alt=\"\" class=\"wp-image-294\" srcset=\"https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/01\/Sol74b.png 600w, https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/01\/Sol74b-295x300.png 295w, https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/01\/Sol74b-147x150.png 147w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/figure>\n\n\n\n<p>\u96e3\u5ea6 5 \u7684\u62fc\u7247\u9664\u53bb\u9084\u672a\u77e5\u7684 11\u300112\u300113\uff0c\u5176\u4ed6\u7247\u7684\u5927\u5c0f\u6211\u5011\u662f\u5df2\u77e5\u7684\uff1b\u800c\u7576\u5b83\u5011\u62fc\u6210\u5716\u4e2d\u9019\u6a23\u6642\uff0c3 \u865f\u62fc\u7247\u7684\u300c1.5\u300d\u55ae\u4f4d\u51f9\u5f27\u7e7c\u7e8c\u5ef6\u4f38\u5230\u76e4\u908a\u5c31\u5207\u51fa 12 \u865f\u62fc\u7247\uff0c\u628a\u9019\u5f27\u7ffb\u8f49\u5c31\u80fd\u5207\u51fa 11 \u865f\u8449\u5b50\u62fc\u7247\uff0c13 \u865f\u4e5f\u5c31\u5207\u51fa\u4f86\u4e86\u3002\u4f46\u662f\u9019\u500b\u5f27\u9577\u537b\u770b\u4f3c\u7121\u6cd5\u4e0b\u624b\u53bb\u6c42\u9577\u5ea6\u2026\u2026<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u5ea7\u6a19\u5316<\/h2>\n\n\n\n<p>\u7d14\u5e7e\u4f55\u65b9\u6cd5\u78b0\u58c1\u7684\u8a71\uff0c\u53ea\u597d\u7528\u4e0a\u4e00\u9ede\u66b4\u529b\u65b9\u6cd5\u6c42\u89e3\u4e86\uff1a\u90a3\u5c31\u662f\u5ea7\u6a19\u5316\u3002\u5982\u679c\u80fd\u6c42\u51fa\u9019\u500b\u5f27\u6240\u5728\u7684\u5713\u61c9\u8a72\u5c31\u80fd\u6c42\u51fa\u9577\u5ea6\u4e86\u3002<\/p>\n\n\n\n<p>\u5c07\u9019\u500b\u5713\u76e4\u64fa\u5230\u5ea7\u6a19\u5e73\u9762\u4e0a\u3002\u4fdd\u6301\u5c0d\u7a31\uff0c\u5713\u76e4\u7684\u5713\u5fc3\u5728\u539f\u9ede\u975e\u5e38\u5408\u7406\uff1b\u4f46\u5ea7\u6a19\u8ef8\u65b9\u5411\u8981\u600e\u9ebc\u8a02\uff1f\u9019\u88e1\u5728\u908a\u908a\u7684 10 \u865f\u62fc\u7247\u7d66\u4e86\u6211\u5011\u65b9\u5411\uff1a\u5b83\u7684\u9760\u5916\u5f27\u662f 3 \u55ae\u4f4d\u5f27 (\\(90\\degree\\))\uff01\u65bc\u662f\u82e5\u5c07 10 \u865f\u62fc\u7247\u7684\u5c16\u89d2\u653e\u5728 \\(A(-1,0)\\)\uff0c\u5247\u5b83\u7684\u920d\u89d2\u5c31\u6703\u5728 \\(B(0,-1)\\)\uff0c\u63a5\u8457\u5c31\u80fd\u7531\u6b64\u6c42\u51fa\u6240\u6709\u4ea4\u9ede\u7684\u5ea7\u6a19\u4e86\u3002\u4e0a\u5716\u53ea\u6709\u628a\u5ea7\u6a19\u8ef8\u7701\u7565\u4e86\uff0c\u4f46\u662f\u5404\u7247\u7684\u4f4d\u7f6e\u662f\u7528\u9019\u500b\u5ea7\u6a19\u756b\u51fa\u4f86\u7684\u3002\u63a5\u4e0b\u4f86\u80fd\u6c42\u7684\u5ea7\u6a19\u4f9d\u5e8f\u662f\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>10 \u865f\u62fc\u7247\u7684\u7b2c\u4e09\u9ede \\(C\\)\uff0c\u9019\u4e5f\u662f\u4e94\u500b\u62fc\u7247\u6240\u805a\u96c6\u7684\u9ede\u3002\u9019\u88e1\u53ef\u4ee5\u7531\u8ddd\u96e2\u5217\u806f\u7acb\u89e3\uff0c\u6216\u662f\u53ef\u4ee5\u5c07 \\(B\\) \u5c0d \\(A\\) \u65cb\u8f49 \\(30\\degree\\) \u5f97\u5230\u3002\u7121\u8ad6\u5982\u4f55\uff0c\u6c42\u5f97\u7684\u5ea7\u6a19\u662f \\(C\\left(\\frac{\\sqrt3-1}2,-\\frac{\\sqrt3-1}2\\right)\\)\u3002<\/li>\n\n\n\n<li>6 \u865f\u62fc\u7247\u7684\u5c16\u9ede \\(D\\)\uff0c\u4e00\u6a23\u5730\u53ef\u4ee5\u7531\u8ddd\u96e2\u5217\u806f\u7acb\u89e3\uff0c\u6216\u662f\u5c07\u539f\u9ede \\(O\\) \u5c0d \\(A\\) \u65cb\u8f49 \\(30\\degree\\)<sup data-fn=\"2b96d1d8-2e81-4d21-8895-76eb95e32392\" class=\"fn\"><a href=\"#2b96d1d8-2e81-4d21-8895-76eb95e32392\" id=\"2b96d1d8-2e81-4d21-8895-76eb95e32392-link\">1<\/a><\/sup>\uff0c\u90fd\u53ef\u5f97\u5230 \\(D\\left(\\frac{\\sqrt3}2-1,\\frac12\\right)\\)\u3002<\/li>\n\n\n\n<li>3 \u865f\u62fc\u7247\u7684\u5c16\u9ede \\(E\\)\uff0c\u9019\u500b\u9ede\u6c92\u6709\u4ec0\u9ebc\u53ef\u4ee5\u8f49\u7684\u57fa\u6e96\uff0c\u6240\u4ee5\u53ea\u597d\u5217\u8ddd\u96e2\u806f\u7acb\u89e3\u3002\u89e3\u51fa\u4f86\u7684\u5ea7\u6a19\u662f\\(E\\left(\\frac{\\sqrt3-1}2,\\frac{\\sqrt3-1}2\\right)\\)\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u5230\u9019\u88e1\u6211\u5011\u8981\u7684\u5f27\u7684\u7aef\u9ede \\(C\\) \u548c \\(E\\) \u5df2\u7d93\u6c42\u51fa\u4f86\u4e86\u3002\u5e78\u904b\u7684\u662f\u9019\u5169\u500b\u9ede\u7684 \\(x\\) \u5ea7\u6a19\u76f8\u7b49\uff0c\\(y\\) \u5ea7\u6a19\u4e92\u70ba\u76f8\u53cd\u6578\uff0c\u56e0\u6b64\u6211\u5011\u8981\u7684\u5713\u5fc3\u5ea7\u6a19\u7684 \\(y\\) \u5ea7\u6a19\u4e00\u5b9a\u662f 0\u3002\\(x\\) \u5ea7\u6a19\u5247\u7531\u7562\u6c0f\u5b9a\u7406\u53ef\u4ee5\u6c42\uff1a\\(\\sqrt{1-(\\frac{\\sqrt3-1}2)^2}=\\frac12\\sqrt{2\\sqrt3}\\)\uff0c\u5373 \\(x\\) \u5ea7\u6a19\u70ba\\(\\frac12(\\sqrt3-1+\\sqrt{2\\sqrt3})\\)\u3002\u4e5f\u5c31\u662f\u8aaa\uff0c\u6211\u5011\u8981\u627e\u7684\u5713\u5fc3\u5728 \\(Q(\\frac12(\\sqrt3-1+\\sqrt{2\\sqrt3}),0)\\)\uff01<\/p>\n\n\n\n<p>\u9019\u500b\u6c42\u51fa\u4f86\u7684 \\(x\\) \u5ea7\u6a19\u9084\u6709\u4e00\u500b\u5f88\u6709\u8da3\u7684\u7279\u6027\uff1a\u4ee4\u5b83\u70ba \\(K_x\\) \u7684\u8a71\uff0c\u5b83\u7684\u5e73\u65b9\u662f $$\\begin{alignat}{1}<br>K_x^2&amp;=\\frac14(4-2\\sqrt3+2(\\sqrt3-1)\\sqrt{2\\sqrt3}+2\\sqrt3)<br>\\\\&amp;=1+\\frac12(\\sqrt3-1)\\sqrt{2\\sqrt3}=1+K_s<br>\\end{alignat}$$ \u4e5f\u5c31\u662f\u6211\u5011\u5176\u5be6\u6709 \\(K_x=\\sqrt{1+K_s}\\)\u3002<\/p>\n\n\n\n<p>\u6709\u4e86\u5713\u5fc3\u5ea7\u6a19\u5c31\u80fd\u6c42\u5169\u500b\u5713\u7684\u4ea4\u9ede\u4e86\u3002\u5713 \\(O\\) \u548c\u5713 \\(Q\\) \u7684 \\(y\\) \u5ea7\u6a19\u76f8\u7b49\uff0c\u6240\u4ee5\u4ea4\u9ede\u7684 \\(x\\) \u5ea7\u6a19\u6703\u548c\u7dda\u6bb5 \\(\\overline{OQ}\\) \u7684\u4e2d\u9ede\u7684 \\(x\\) \u5ea7\u6a19\u76f8\u7b49\uff0c\u662f \\(\\frac12K_x\\)\u3002\\(y\\) \u5ea7\u6a19\u5c31\u518d\u5ea6\u4f7f\u7528\u7562\u6c0f\u5b9a\u7406\u6c42\u4e4b\uff1a\\(1-(\\frac12K_x)^2=1-\\frac14(1+K_s)=\\frac34-\\frac14K_s=\\frac14(3-K_s)\\) \u3002\u65bc\u662f\u6211\u5011\u7684\u4ea4\u9ede\u5ea7\u6a19\u5373\u70ba \\(F(\\frac12K_x,-\\frac12\\sqrt{3-K_s})\\)\u3002<\/p>\n\n\n\n<p>\u4ee5\u4e0b\u518d\u8a18\\(K_y=\\sqrt{3-K_s}=\\sqrt{3-\\sqrt{2\\sqrt3-3}}\\)\uff0c\u4e5f\u5c31\u662f \\(F\\) \u9ede\u5ea7\u6a19\u5c31\u662f\\((\\frac12K_x,-\\frac12K_y)\\)\u3002\u9019\u5169\u500b\u503c\u7684\u5f62\u5f0f\u4e0a\u53ef\u4ee5\u770b\u5230\u6709 \\(K_x^2+K_y^2=4\\)\uff0c\u9019\u548c \\(F\\) \u9ede\u5230\u539f\u9ede \\(O\\) \u7684\u9577\u5ea6\u70ba 1 \u543b\u5408\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">11 \u865f\u62fc\u7247<\/h2>\n\n\n\n<p>\u4ea4\u9ede\u7b97\u51fa\u4f86\u4e86\uff0c\u63a5\u4e0b\u4f86\u8981\u4f86\u6c42\u5f27 \\(\\overarc{EF}\\) \u7684\u5f27\u89d2 \\(\\beta\\)\u3002<br>\u7528\u7684\u65b9\u6cd5\u662f\u5411\u91cf\u5167\u7a4d\uff0c\u7531\u5713\u5fc3\u9023\u5411\u9019\u5169\u500b\u9ede\u7684\u55ae\u4f4d\u5411\u91cf\u5167\u7a4d\u5373\u662f\u5b83\u5011\u7684\u593e\u89d2\u4e86\uff1a$$\\begin{alignat}{1}<br>\\vec{QE}&amp;=\\left(-\\frac{\\sqrt{2\\sqrt3}}2,\\frac{\\sqrt3-1}2\\right)<br>\\\\\\vec{QF}&amp;=\\left(-\\frac12K_x,-\\frac12K_y\\right)<br>\\\\\\cos\\beta&amp;=\\vec{QE}\\cdot\\vec{QF}<br>\\\\&amp;=\\frac{\\sqrt{2\\sqrt3}}2\\cdot\\frac12K_x-\\frac{\\sqrt3-1}2\\cdot\\frac12K_y<br>\\\\&amp;=\\frac{\\sqrt{2\\sqrt3}}2\\cdot\\frac14(\\sqrt3-1+\\sqrt{2\\sqrt3})-\\frac{\\sqrt3-1}2\\cdot\\frac12K_y<br>\\\\&amp;=\\frac18\\left((\\sqrt3-1)\\sqrt{2\\sqrt3}+2\\sqrt3\\right)-\\frac14(\\sqrt3-1)K_y<br>\\\\&amp;=\\frac14K_s+\\frac14\\sqrt3-\\frac14(\\sqrt3-1)K_y<br>\\\\&amp;=\\frac14(K_s+\\sqrt3-(\\sqrt3-1)K_y)<br>\\end{alignat}$$ \u70ba\u4e86\u5f8c\u7e8c\u8a08\u7b97\uff0c\u5148\u628a\u9019\u500b\u89d2\u7684\u6b63\u5f26\u503c\u4e5f\u6c42\u51fa\u4f86\u3002\u8a08\u7b97\u6709\u4e00\u9ede\u7e41\u6240\u4ee5\u76f4\u63a5\u4e0a\u7d50\u679c\uff1a<br>$$\\begin{alignat}{1}<br>\\cos^2\\beta=-\\frac12(\\sqrt3-1)(K_s+\\sqrt3)(K_y-2)<br>\\\\\\sin^2\\beta=1+\\frac12(\\sqrt3-1)(K_s+\\sqrt3)(K_y-2)<br>\\\\\\sin\\beta=\\sqrt{1+\\frac12(\\sqrt3-1)(K_s+\\sqrt3)(K_y-2)}<br>\\end{alignat}$$ \u611f\u89ba\u4e0d\u50cf\u662f\u500b\u6709\u597d\u8868\u793a\u5f0f\u7684\u6771\u897f\u3002<br>\u9019\u500b\u5f27\u89d2\u6709\u591a\u5927\u5462\uff1fMathematica \u544a\u8a34\u6211\u7684\u8fd1\u4f3c\u503c\u662f \\(\\beta\\approx2.36853\\times\\frac\\pi6\\)\u3002<br>11 \u865f\u7684\u9762\u7a4d\u7528\u9019\u500b\u5f27\u89d2\u8868\u793a\u5f88\u7c21\u55ae\uff1a\\(2\\seg(\\beta)=\\beta-\\sin\\beta\\)\uff0c\u4f46\u662f\u4ee3\u9032\u53bb\u5c31\u662f\u66f4\u5927\u4e00\u5718\u7684\u6d88\u4e0d\u6389\u7684\u6771\u897f\uff1a$$<br>A_{11}=\\beta-\\sin\\beta=\\arccos\\left(\\frac14(K_s+\\sqrt3-(\\sqrt3-1)K_y)\\right)-\\sqrt{1+\\frac12(\\sqrt3-1)(K_s+\\sqrt3)(K_y-2)}<br>$$ \u8fd1\u4f3c\u503c\u662f \\(A_{11}\\approx0.294325\\)\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">13 \u865f\u62fc\u7247<\/h2>\n\n\n\n<p>\u63a5\u4e0b\u4f86\u8f2a\u5230 13 \u865f\u62fc\u7247\u4e86\uff1b\u9996\u5148\u5148\u6c42\u5b83\u7684\u9577\u5f27\u89d2 \\(\\overarc{FG}=\\alpha\\)\u3002\\(G\\) \u9ede\u5ea7\u6a19\u5176\u5be6\u5f88\u597d\u6c42\uff1a\u5f9e\u5728\u8ca0 x \u8ef8\u4e0a\u7684 \\(A\\) \u9ede\u958b\u59cb\uff0c\u7d93\u904e 5-2-7 \u865f\u7684\u5916\u570d\u5230\u9054 \\(G\\) \u9ede\u30025-2 \u865f\u5f27\u9577 4 \u55ae\u4f4d\uff0c7 \u865f\u9019\u500b\u908a\u662f 1 \u55ae\u4f4d\uff0c\u6240\u4ee5 \\(G\\) \u9ede\u7684\u5e45\u89d2\u662f\u6c34\u5e73\u8d77\u7b97\u6b63 1 \u55ae\u4f4d\uff0c\u4e5f\u5c31\u662f\u5ea7\u6a19\u662f \\(G\\left(\\frac{\\sqrt3}2,\\frac12\\right)\\)\u3002\u65bc\u662f\u6211\u5011\u8981\u6c42\u7684 \\(\\alpha\\) \u5373\u70ba \\(\\frac\\pi6\\) \u6e1b\u53bb \\(F\\) \u7684\u5e45\u89d2 \\(\\arcsin(-\\frac12K_y)\\)\uff0d\uff0d\u540c\u6a23\u7528\u500d\u89d2\u516c\u5f0f\u6d88\u6389 \\(K_y\\) \u6700\u5916\u5c64\u7684\u6839\u865f\uff1a$$\\begin{alignat}{1}<br>\\xi&amp;=\\arcsin(-\\frac12K_y)\\\\<br>\\sin^2\\xi&amp;=\\frac14(3-K_s)\\\\<br>\\cos2\\xi&amp;=1-2\\sin^2\\xi=\\frac12(K_s-1)\\\\<br>\\xi&amp;=\\pm\\frac12\\arccos\\left(\\frac12(K_s-1)\\right) \\mathtt{(\u6b63\u4e0d\u5408)}\\\\<br>\\alpha&amp;=\\frac\\pi6-\\xi=\\frac\\pi6+\\frac12\\arccos\\left(\\frac12(K_s-1)\\right)\\approx2.65284\\times\\frac\\pi6<br>\\end{alignat}$$ \u7576\u5e74\u6211\u7684\u76f4\u63a5\u6bd4\u8f03\u662f\u8aaa\u5b83\u8ddf 2 \u865f\u62fc\u7247\u7684\u300c2.5\u300d\u55ae\u4f4d\u5dee\u4e0d\u591a\uff0c\u5be6\u969b\u6578\u5b57\u4e5f\u78ba\u8b49\u9019\u4e00\u9ede\uff1a2 \u865f\u62fc\u7247\u7684\u300c2.5\u300d\u55ae\u4f4d\u5be6\u969b\u4e0a\u662f 4 &#8211; 1.43138 = 2.56862\u55ae\u4f4d\u5f27\uff0c13 \u865f\u62fc\u7247\u7684\u9577\u5f27\u53c8\u7a0d\u5fae\u9577\u4e00\u4e9b\u4e9b\u4e86\u3002<\/p>\n\n\n\n<p>\u63a5\u4e0b\u4f86\u662f\u9762\u7a4d\u300213 \u865f\u62fc\u7247\u7684\u4e09\u908a\u4e2d\uff0c\u77ed\u908a\u662f 1 \u55ae\u4f4d\u5f27\uff0c\u6545\u5f26\u9577\u662f \\(\\frac12(\\sqrt6-\\sqrt2)\\)\uff1b\u9760 11 \u7684\u5167\u5f27\u662f \\(\\beta\\) \u89d2\uff0c\u5f26\u9577\u662f \\(2\\sin\\frac\\beta2\\)\uff1b\u9760\u5916\u7684\u5916\u5f27\u662f \\(\\alpha\\) \u89d2\uff0c\u5f26\u9577\u662f \\(2\\sin\\frac\\alpha2\\)\u3002<br>\u5148\u6c42 \\(2\\sin\\frac\\beta2\\)\uff0c\u9019\u500b\u53ef\u4ee5\u5efa\u7acb\u5728\u4e0a\u9762\u6c42\u904e\u7684 \\(\\cos\\beta\\) \u4e0a\uff1a$$\\begin{alignat}{1}<br>2\\sin\\frac\\beta2&amp;=2\\sqrt{\\frac{1-\\cos\\beta}2}\\\\<br>&amp;=\\sqrt{2-2\\cos\\beta}\\\\<br>&amp;=\\sqrt{2+\\frac12(K_y^2+(\\sqrt3-1)K_y-(3+\\sqrt3))}\\\\<br>&amp;=\\sqrt{\\frac12K_y^2+\\frac{\\sqrt3-1}2K_y+\\frac12(1-\\sqrt3)}\\\\<br>&amp;=\\sqrt{\\frac12\\left(K_y+\\frac{\\sqrt3-1}{2}\\right)^2-\\frac{\\sqrt3}4}<br>\\end{alignat}$$ \\(2\\sin\\frac\\alpha2\\) \u5247\u4e00\u6a23\u5148\u6c42 \\(\\cos\\alpha\\)\uff1a$$\\begin{alignat}{1}<br>\\cos\\alpha&amp;=\\cos\\left(\\frac\\pi6+\\arcsin\\left(\\frac12K_y\\right)\\right)\\\\<br>&amp;=\\frac{\\sqrt3}2\\cos\\arcsin\\left(\\frac12K_y\\right)-\\frac12\\sin\\arcsin\\left(\\frac12K_y\\right)\\\\<br>&amp;=\\frac{\\sqrt3}2\\sqrt{1-\\left(\\frac12K_y\\right)^2}-\\frac12\\cdot\\frac12K_y\\\\<br>&amp;=\\frac{\\sqrt3}4\\sqrt{4-K_y^2}-\\frac14K_y\\\\<br>&amp;=\\frac{\\sqrt3}4K_x-\\frac14K_y\\\\<br>&amp;=\\frac14(\\sqrt3K_x-K_y)\\\\<br>2\\sin\\frac\\alpha2&amp;=\\sqrt{2-2\\cos\\alpha}\\\\<br>&amp;=\\sqrt{2-\\frac{\\sqrt3}2K_x+\\frac12K_y}<br>\\end{alignat}$$<br>\u5f9e\u9019\u88e1\u8981\u6c42\u51fa\u4efb\u4f55\u4e00\u500b\u89d2\u5ea6\u90fd\u662f\u4e00\u500b\u6d69\u5927\u7684\u5de5\u7a0b\u3002\u4e0d\u904e\u773c\u5c16\u7684\u4eba\u53ef\u80fd\u770b\u5230\u4e86\u4e0a\u5716 13 \u865f\u62fc\u7247\u7684\u57fa\u5e95\u4e09\u89d2\u5f62\u597d\u50cf\u6709\u500b\u7384\u6a5f\u2026\u2026<\/p>\n\n\n\n<p>\u90a3\u5c31\u662f \\(\\triangle EFG\\) \u597d\u50cf\u662f\u500b\u76f4\u89d2\u4e09\u89d2\u5f62\uff1f\uff01\u4f86\u5be6\u969b\u8a08\u7b97\u770b\u770b\u5427\uff1a$$\\begin{alignat}{1}<br>\\overline{EG}^2+\\overline{FG}^2&amp;=\\left(\\frac12(\\sqrt6-\\sqrt2)\\right)^2+\\left(\\sqrt{\\frac12(K_y+\\frac{\\sqrt3-1}{2})^2-\\frac{\\sqrt3}4}\\right)^2\\\\<br>&amp;=(2-\\sqrt3)+\\left(\\frac12\\left(K_y+\\frac{\\sqrt3-1}{2}\\right)^2-\\frac{\\sqrt3}4\\right)\\\\<br>&amp;=\\frac12\\left(K_y+\\frac{\\sqrt3-1}{2}\\right)^2+2-\\frac54\\sqrt3\\\\<br>\\overline{EF}^2&amp;=\\left(\\sqrt{2-\\frac{\\sqrt3}2K_x+\\frac12K_y}\\right)^2\\\\<br>&amp;=2-\\frac{\\sqrt3}2K_x+\\frac12K_y<br>\\end{alignat}$$\u2026\u2026\u770b\u8d77\u4f86\u4e0d\u50cf\u76f8\u7b49\u3002\u4ee5 Mathematica \u8a08\u7b97\u53ef\u77e5\u9019\u5169\u500b\u7d50\u679c\u5dee\u4e86\u7d04 1.2%\uff0c\u6240\u4ee5\u90a3\u500b\u89d2\u4e26\u4e0d\u662f\u76f4\u89d2\uff0c\u5b83\u5927\u7d04\u662f \\(90.94\\degree\\)\u3002<\/p>\n\n\n\n<p>\u65e2\u7136\u6c92\u6709\u7279\u6b8a\u89d2\u53ea\u597d\u8001\u8001\u5be6\u5be6\u56de\u5230\u6d77\u9f8d\u516c\u5f0f\u4f86\u6c42\u9762\u7a4d\uff0d\uff0d\u6216\u8005\u56e0\u70ba\u9019\u5e7e\u500b\u908a\u9577\u90fd\u6709\u6839\u865f\uff0c\u7528\u79e6\u4e5d\u97f6\u516c\u5f0f\u6216\u8a31\u6703\u597d\u7b97\u4e00\u9ede\uff1a\u5728$$\\triangle_{13}=\\frac12\\sqrt{a^2c^2-\\left(\\frac{a^2+c^2-b^2}2\\right)^2}$$ \u7576\u4e2d\u4ee4 $$\\begin{cases}<br>a^2=2-\\frac{\\sqrt3}2K_x+\\frac12K_y\\\\<br>c^2=\\frac12\\left(K_y+\\frac{\\sqrt3-1}{2}\\right)^2-\\frac{\\sqrt3}4\\\\<br>b^2=2-\\sqrt3\\end{cases}$$ \u4ee3\u5165\u2026\u2026\u7136\u5f8c\u5c31\u767c\u73fe\u6839\u865f\u88e1\u9084\u662f\u4e00\u5927\u5718\u3002Mathematica \u628a \\(\\triangle_{13}\\) \u5316\u7c21\u5beb\u6210\u6839\u5f0f\u4e4b\u5f8c\u662f\u9577\u9019\u6a23\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"843\" height=\"160\" src=\"https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/02\/Triangle13expression.png\" alt=\"\" class=\"wp-image-420\" srcset=\"https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/02\/Triangle13expression.png 843w, https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/02\/Triangle13expression-300x57.png 300w, https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/02\/Triangle13expression-150x28.png 150w, https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/02\/Triangle13expression-768x146.png 768w\" sizes=\"auto, (max-width: 843px) 100vw, 843px\" \/><\/figure>\n\n\n\n<p>\u2026\u2026\u5c31\u64fa\u8457\u597d\u4e86\u3002<br>\u5982\u679c\u7b97\u4e0a\u5f13\u5f62\u9762\u7a4d\u72c0\u6cc1\u4e26\u4e0d\u6703\u66f4\u597d\uff0c\u56e0\u70ba\u9664\u4e86\u77ed\u51f9\u5f27\u6709\u77ed\u4e00\u9ede\u7684\u5f0f\u5b50 \\(\\frac\\pi{12}-\\frac14\\) \u4e4b\u5916\uff0c\u9577\u51f9\u5f27 \\(\\frac12(\\beta-\\sin\\beta)\\) \u548c\u9577\u51f8\u5f27 \\(\\frac12(\\alpha-\\sin\\alpha)\\) \u90fd\u4e0d\u662f\u597d\u7b97\u7684\u6771\u897f\uff0c\u800c\u4e14\u6703\u8ddf 11 \u865f\u62fc\u7247\u4e00\u6a23\u6703\u7522\u751f\u6d88\u4e0d\u6389\u7684\u53cd\u4e09\u89d2\u51fd\u6578\uff0c\u6240\u4ee5\u5c31\u53ea\u80fd\u9019\u6a23\u5217\u5f0f\u51fa\u4f86\u8868\u793a\u4e86\uff1a$$A_{13}=\\triangle_{13}-\\left(\\frac\\pi{12}-\\frac14\\right)+\\frac12\\left(\\alpha-\\beta-\\sin\\alpha+\\sin\\beta\\right)\\approx0.344548$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">12 \u865f\u7684\u5f27\u89d2\u548c\u9762\u7a4d<\/h2>\n\n\n\n<p>\u6700\u5f8c\u662f\u5c0f\u584a\u7684 12 \u865f\u4e86\u3002\u5b83\u548c 10 \u865f\u63a5\u89f8\u7684\u9577\u51f9\u5f27\u662f 10 \u865f\u7684\u77ed\u51f8\u5f27\u300c1.5\u300d\u55ae\u4f4d\uff0c\u800c\u6709\u4e86\u4e4b\u524d\u6c42\u7684 \\(\\alpha,\\beta\\) \u89d2\uff0c\u53e6\u5916\u5169\u500b\u5f27\u4e5f\u53ef\u4ee5\u5beb\u51fa\u4f86\uff1a\u548c 11 \u865f\u63a5\u89f8\u7684\u5c0f\u51f9\u5f27\u662f \\(\\beta-\\arccos(\\sqrt3-1)\\approx0.937154\\times\\frac\\pi6\\)\uff0c\u548c\u5916\u5708\u63a5\u89f8\u7684\u51f8\u5f27\u662f \\(\\frac{2\\pi}3-\\alpha\\approx1.34716\\times\\frac\\pi6\\)\u3002\u7576\u5e74\u6211\u7528\u7d19\u7247\u76f4\u63a5\u6bd4\u8f03\u7684\u7d50\u679c\u662f\uff1a\u51f8\u300c1.5\u300d\u3001\u51f9 1\u3001\u51f9\u300c1.5\u300d\uff0c\u548c\u5be6\u969b\u6578\u5b57\u78ba\u5be6\u5dee\u4e0d\u591a\u3002<\/p>\n\n\n\n<p>\u9762\u7a4d\u65b9\u9762\uff0c\u9019\u88e1\u6709\u4e00\u500b\u4e0d\u7528\u518d\u53bb\u6c42\u8907\u96dc\u5f27\u9577\u7684\u65b9\u5f0f\uff1a\u56de\u5230\u7576\u5e74\u7684\u7b2c\u4e00\u7bc7\u6587\u7ae0\uff0c\u5e15\u7d22\u5728\u63a8\u6587\u88e1\u6709\u554f\u8aaa\uff1a<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u2192 puzzlez:\u90a3\u8acb\u554f 11 -12 -13 \u53ef\u4e0d\u53ef\u4ee5\u62fc\u51fa 8 \uff1f<\/p>\n<\/blockquote>\n\n\n\n<p>\u800c\u6211\u5728\u7b2c\u4e09\u7bc7\u6587\u7ae0\u88e1\u9019\u9ebc\u56de\u7b54\u4e86\uff1a<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u62ff\u7d19\u5207\u7684\u62fc\u7247\u62fc\u4e00\u62fc\u5f8c\u61c9\u8a72\u662f 2,3,5,6,7,10,11,12,13<sup data-fn=\"4079b66d-0e5a-40f3-a43f-cc1c6cfb3d7f\" class=\"fn\"><a href=\"#4079b66d-0e5a-40f3-a43f-cc1c6cfb3d7f\" id=\"4079b66d-0e5a-40f3-a43f-cc1c6cfb3d7f-link\">2<\/a><\/sup><br>\u5e15\u7d22\u4f60\u554f\u7684 11~13 \u662f\u4e0d\u662f\u80fd\u62fc\u6210 8 \u865f\u61c9\u8a72\u662f\u5f9e\u9019\u88e1\u4f86\u7684<br>\u56e0\u70ba 2,3,5,6,7,8,10<sup data-fn=\"e9ab6017-a0c8-498b-8fb3-5d8d3c0209b0\" class=\"fn\"><a href=\"#e9ab6017-a0c8-498b-8fb3-5d8d3c0209b0\" id=\"e9ab6017-a0c8-498b-8fb3-5d8d3c0209b0-link\">3<\/a><\/sup> \u4e5f\u53ef\u4ee5\u62fc\u8d77\u4f86&#8230;<br>\u4e0d\u904e\u5982\u679c\u80fd\u62fc\u51fa 8 \u865f\u6211\u90a3\u984c 10-3 \u5c31\u89e3\u958b\u4e86<br>\u6240\u4ee5\u61c9\u8a72\u662f\u4e0d\u80fd\u62fc\u4f46\u9762\u7a4d\u7e3d\u548c\u76f8\u7b49\u2026<\/p>\n<\/blockquote>\n\n\n\n<p>\u4e5f\u5c31\u662f\u8aaa\uff0c\\(A_{11}+A_{12}+A_{13}=A_8=\\frac\\pi4\\)\u3002\u8a73\u7d30\u5f0f\u5b50\u53ef\u4ee5\u7531\u4e0a\u9762\u8a08\u7b97\u7d50\u679c\u5957\u7528\u800c\u5f97\u6240\u4ee5\u9019\u88e1\u4e5f\u4e0d\u5beb\u4e86\uff0c\u53ea\u7d66\u51fa\u9762\u7a4d\u8fd1\u4f3c\u503c \\(A_{12}\\approx0.146526\\)\u3002\u9019\u548c\u7576\u5e74\u6211\u7d66\u51fa\u4f86\u7684 \\(0.099865\\) \u4e4b\u9593\u7684\u8aa4\u5dee\u5c31\u662f\u5728\u4e0a\u9762\u91cf\u7684\u5f27\u9577\u7684\u5dee\u5225 \uff0c\u5c24\u5176\u662f\u9577\u51f8\u5f27\u5dee\u4e86 10% \u5de6\u53f3\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u7e3d\u6574\u7406\u548c\u4e00\u500b\u6709\u8da3\u7684\u554f\u984c<\/h2>\n\n\n\n<p>\u81f3\u6b64\u6240\u6709\u62fc\u7247\u7684\u9762\u7a4d\u90fd\u5df2\u7d93\u6c42\u51fa\u4f86\u4e86\uff0c\u9019\u88e1\u5c07\u5168\u90e8 14 \u7247\u7684\u9762\u7a4d\u6574\u7406\u6210\u8868\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">\u62fc\u7247<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u9762\u7a4d\u8fd1\u4f3c\u503c (\u5168\u5713\u76e4\u70ba \\(\\pi\\approx3.141593\\))<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u9762\u7a4d\u4f54\u5168\u5713\u76e4\u7684\u6bd4\u4f8b<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.238201<\/td><td class=\"has-text-align-center\" data-align=\"center\">7.5821%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.338931<\/td><td class=\"has-text-align-center\" data-align=\"center\">10.7885%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.227690<\/td><td class=\"has-text-align-center\" data-align=\"center\">7.2476%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.295909<\/td><td class=\"has-text-align-center\" data-align=\"center\">9.4191%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.331730<\/td><td class=\"has-text-align-center\" data-align=\"center\">10.5593%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.604226<\/td><td class=\"has-text-align-center\" data-align=\"center\">19.2331%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">7<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.319508<\/td><td class=\"has-text-align-center\" data-align=\"center\">10.1703%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.785398<\/td><td class=\"has-text-align-center\" data-align=\"center\">25%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.251289<\/td><td class=\"has-text-align-center\" data-align=\"center\">7.9988%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">10<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.534109<\/td><td class=\"has-text-align-center\" data-align=\"center\">17.0012%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">11<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.294325<\/td><td class=\"has-text-align-center\" data-align=\"center\">9.3687%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">12<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.146526<\/td><td class=\"has-text-align-center\" data-align=\"center\">4.6641%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">13<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.344548<\/td><td class=\"has-text-align-center\" data-align=\"center\">10.9673%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">14<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.342427<\/td><td class=\"has-text-align-center\" data-align=\"center\">10.8998%<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>\u6700\u5f8c\uff0c\u6709\u4e86\u9019\u500b\u53ef\u4ee5\u4f86\u56de\u7b54\u4e00\u500b\u6709\u9ede\u6709\u8da3\u7684\u554f\u984c\uff1a\u9f8d\u535a\u58eb\u5713\u76e4\u9084\u6709\u6c92\u6709\u5176\u4ed6\u7684\u7d44\u5408\u53ef\u80fd\u6709\u89e3\uff1f<\/p>\n\n\n\n<p>\u70ba\u6b64\uff0c\u9019\u88e1\u7528\u53e6\u4e00\u7a2e\u65b9\u5f0f\u4f86\u6574\u7406\u9019 14 \u7247\u62fc\u7247\u7684\u9762\u7a4d\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">\u62fc\u7247<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u9762\u7a4d<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac\\pi{12}\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac12\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac{\\sqrt3}2\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\seg(\\theta_s)\\)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac{\\sqrt3}2-\\frac\\pi{12}\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">-1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac\\pi4\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">14<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac{\\sqrt3}2-\\frac\\pi6\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac\\pi{12}-\\seg(\\theta_s)\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">-1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac\\pi{12}+\\seg(\\theta_s)\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">7<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac\\pi4-\\frac12+\\seg(\\theta_s)\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">-1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac\\pi4-\\frac12-\\seg(\\theta_s)\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">-1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">-1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac12-\\frac\\pi{12}\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">-1<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">10<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac12+\\seg(\\theta_s)\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">2+5<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac\\pi2-\\frac{\\sqrt3}2-\\seg(\\theta_s)\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">-1<\/td><td class=\"has-text-align-center\" data-align=\"center\">-1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">11+12+13<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(\\frac\\pi4\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>\u9019\u5f35\u8868\u7684\u53f3\u908a\u53ef\u4ee5\u63d0\u4f9b\u6e4a\u9762\u7a4d\u7684\u53c3\u8003\uff1b\u6240\u9078\u7684\u5217\u6700\u53f3\u908a\u4e09\u6b04\u7684\u503c\u7e3d\u548c\u8981\u70ba 0\u3002\u82e5\u518d\u4f9d\u64da \\(\\frac\\pi{12}\\) \u7684\u6578\u91cf\u5206\u985e\u7684\u8a71\uff0c\u4e00\u5171\u6709\u4ee5\u4e0b\u7d44\u5408\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\(\\frac2{12}\\pi=\\frac16\\pi\\)\uff1a(3,4)<\/li>\n\n\n\n<li>\\(\\frac3{12}\\pi=\\frac14\\pi\\)\uff1a(1,3,7)\u3001(1,4,9)\u3001(8)\u3001(9,10)\u3001(11,12,13)<\/li>\n\n\n\n<li>\\(\\frac5{12}\\pi\\)\uff1a(2,4,5,14) <\/li>\n\n\n\n<li>\\(\\frac6{12}\\pi=\\frac12\\pi\\)\uff1a(1,2,5,7,14)\u3001(2,4,5,6)<\/li>\n\n\n\n<li>\\(\\frac7{12}\\pi\\)\uff1a(1,2,5,6,7)<\/li>\n\n\n\n<li>\\(\\frac8{12}\\pi=\\frac23\\pi\\)\uff1a(2,3,5,7,10,14)<\/li>\n\n\n\n<li>\\(\\frac9{12}\\pi=\\frac34\\pi\\)\uff1a(2,3,5,6,7,10)<\/li>\n<\/ul>\n\n\n\n<p>\u4e0a\u9762\u7684\u7d44\u5408\u4e2d\u53ef\u7531\u5c0f\u7684\u7d44\u5408\u6e4a\u51fa\u4f86\u7684\u5c31\u7565\u53bb\uff0c\u4f8b\u5982\u56db\u5206\u4e4b\u4e09\u5713\u6709\u4e00\u500b (1,2,3,4,5,6,7) \u53ef\u4ee5\u770b\u6210 (1,3,7) + (2,4,5,6) \u6216 (1,2,5,6,7) + (3,4) \u7684\u7d44\u5408\uff0c\u56e0\u6b64\u4e0d\u5217\u5165\u4e0a\u8868\u3002<br>\u5f9e\u9019\u88e1\u518d\u53bb\u7d44\u5408\u51fa\u6574\u5713\u7684\u7d44\u5408\u5171\u6709 12 \u7a2e\uff0c\u53ef\u4ee5\u5206\u6210\u4ee5\u4e0b\u4e09\u985e\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>{(1,2,5,7,14), (2,4,5,6)} + {(1,3,7), (8), (9,10), (11,12,13)}x2\uff1a\u534a\u5713\u4e8c\u9078\u4e00\u52a0\u4e0a\u56db\u5206\u4e4b\u4e00\u5713\u56db\u9078\u4e8c\u3002\u6263\u6389\u91cd\u8986\u62fc\u7247\u7684\u8a71\u5171\u6709\u4e5d\u7a2e\u7d44\u5408\uff1a\n<ul class=\"wp-block-list\">\n<li>(2,4,5,6) + (1,3,7) + (8)\uff1a\u7b2c\u4e8c\u7a2e\u56f0\u96e3\u5ea6<\/li>\n\n\n\n<li>(2,4,5,6) + (1,3,7) + (9,10)\uff1a\u7b2c\u4e00\u7a2e\u56f0\u96e3\u5ea6<\/li>\n\n\n\n<li>(2,4,5,6) + (1,3,7) + (11,12,13)\uff1a\u7b2c\u516b\u7a2e\u56f0\u96e3\u5ea6<sup data-fn=\"35237a18-b815-4044-8535-d386fb66f7ca\" class=\"fn\"><a href=\"#35237a18-b815-4044-8535-d386fb66f7ca\" id=\"35237a18-b815-4044-8535-d386fb66f7ca-link\">4<\/a><\/sup><\/li>\n\n\n\n<li>(2,4,5,6) + (8) + (9,10)\uff1a\u7b2c\u516d\u7a2e\u56f0\u96e3\u5ea6<\/li>\n\n\n\n<li>(2,4,5,6) + (8) + (11,12,13)\uff1a???<\/li>\n\n\n\n<li>(2,4,5,6) + (9,10) + (11,12,13)\uff1a???<\/li>\n\n\n\n<li>(1,2,5,7,14) + (8) + (9,10)\uff1a\u7b2c\u4e09\u7a2e\u56f0\u96e3\u5ea6<\/li>\n\n\n\n<li>(1,2,5,7,14) + (8) + (11,12,13)\uff1a???<\/li>\n\n\n\n<li>(1,2,5,7,14) + (9,10) + (11,12,13)\uff1a\u7b2c\u4e5d\u7a2e\u56f0\u96e3\u5ea6<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>(2,3,5,6,7,10) + {(8), (11,12,13)}\uff1a\u56db\u5206\u4e4b\u4e09\u5713\u52a0\u4e0a\u56db\u5206\u4e4b\u4e00\u5713\u4e8c\u9078\u4e00\u3002\u9019\u5206\u5225\u662f\uff1a\n<ul class=\"wp-block-list\">\n<li>(2,3,5,6,7,10) + (8)\uff1a\u7b2c\u56db\u7a2e\u56f0\u96e3\u5ea6<\/li>\n\n\n\n<li>(2,3,5,6,7,10) + (11,12,13)\uff1a\u7b2c\u4e94\u7a2e\u56f0\u96e3\u5ea6<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>(1,3,7) + (8) + (9,10) + (11,12,13)\uff1a\u56db\u500b\u56db\u5206\u4e4b\u4e00\u5713\u5168\u9078\u3002\u9019\u662f\u7b2c\u4e03\u7a2e\u56f0\u96e3\u5ea6\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u7531\u65bc\u62fc\u7247\u8a2d\u8a08\u7684\u95dc\u4fc2\uff0c\u96d6\u7136\u5b58\u5728\u80fd\u5920\u7d44\u6210\u9762\u7a4d\u5c0d\u7684\u7b54\u6848\u4f46\u4e26\u4e0d\u80fd\u76f4\u63a5\u80af\u5b9a\u6709\u6c92\u6709\u89e3\uff1b\u4e0d\u7136\u7576\u5e74\u90a3\u500b\u7121\u89e3\u984c\u5c31\u4e0d\u6703\u7121\u89e3\u4e86 (\u82e6\u7b11)\u3002\u6240\u4ee5\u5269\u4e0b\u7684\u4e09\u500b\u7d44\u5408 (2,4,5,6,8,11,12,13)\u3001(2,4,5,6,9,10,11,12,13)\u3001(1,2,5,7,8,11,12,13,14) \u7a76\u7adf\u6709\u6c92\u6709\u89e3\u53ef\u80fd\u4e5f\u8981\u591a\u65b9\u5617\u8a66\u624d\u80fd\u77e5\u9053\u4e86\u3002<\/p>\n\n\n\n<p>\u90a3\u9ebc\u9019\u5c31\u662f\u5c0d\u7576\u5e74\u7684\u6311\u6230\u984c\uff1a\u5168\u90e8\u5341\u56db\u7247\u62fc\u7247\u7684\u9762\u7a4d\u7684\u8a73\u7d30\u89e3\u7b54\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u8a3b\u8173<\/h2>\n\n\n<ol class=\"wp-block-footnotes\"><li id=\"2b96d1d8-2e81-4d21-8895-76eb95e32392\">\u7406\u7531\u662f\uff1a\u9996\u5148 \\(\\overline{AD}\\) \u662f 6 \u865f\u62fc\u7247\u7684\u8170\uff0c\u9577\u5ea6\u70ba 1\uff1b\u5176\u659c\u7387\u53ef\u81ea \\(\\overline{AB}\\) \u659c\u7387 \\(-45\\degree\\) \u8d77\u7b97\uff0c\u52a0\u4e0a 10 \u865f\u62fc\u7247\u7684\u9802\u89d2 \\(\\angle BAC = 30\\degree\\)\uff0c\u52a0\u4e0a 6 \u865f\u62fc\u7247\u7684\u5e95\u89d2 \\(\\angle CAD = 45\\degree\\)\uff0c\u7e3d\u8a08\u662f \\(+30\\degree\\)\u3002 <a href=\"#2b96d1d8-2e81-4d21-8895-76eb95e32392-link\" aria-label=\"\u8fd4\u56de\u8a3b\u8173\u53c3\u7167 1\">\u21a9\ufe0e<\/a><\/li><li id=\"4079b66d-0e5a-40f3-a43f-cc1c6cfb3d7f\">\u9019\u5c31\u662f\u9019\u7bc7\u958b\u982d\u63d0\u5230\u7684\u7b2c\u4e94\u7a2e\u56f0\u96e3\u5ea6\u3002 <a href=\"#4079b66d-0e5a-40f3-a43f-cc1c6cfb3d7f-link\" aria-label=\"\u8fd4\u56de\u8a3b\u8173\u53c3\u7167 2\">\u21a9\ufe0e<\/a><\/li><li id=\"e9ab6017-a0c8-498b-8fb3-5d8d3c0209b0\">\u9019\u662f\u7b2c\u56db\u7a2e\u56f0\u96e3\u5ea6\u3002 <a href=\"#e9ab6017-a0c8-498b-8fb3-5d8d3c0209b0-link\" aria-label=\"\u8fd4\u56de\u8a3b\u8173\u53c3\u7167 3\">\u21a9\ufe0e<\/a><\/li><li id=\"35237a18-b815-4044-8535-d386fb66f7ca\">\u4e03\u516b\u4e5d\u9019\u4e09\u7a2e\u56f0\u96e3\u5ea6\u7684\u8cc7\u6599\u662f\u4f86\u81ea <a href=\"https:\/\/www.ptt.cc\/bbs\/puzzle\/M.1312721580.A.B49.html\">playertsai \u7684\u6587\u7ae0<\/a>\u3002 <a href=\"#35237a18-b815-4044-8535-d386fb66f7ca-link\" aria-label=\"\u8fd4\u56de\u8a3b\u8173\u53c3\u7167 4\">\u21a9\ufe0e<\/a><\/li><\/ol>","protected":false},"excerpt":{"rendered":"<p>\u7cfb\u5217\u6587\u7684\u7b2c\u4e09\u7bc7\uff0c\u4f86\u6311\u6230\u5927\u9b54\u738b 11-12-13 \u7d44\u5408\u4e86\u3002\u524d\u9762\u5169\u7bc7\u6587\u7ae0\u53ef\u4ee5\u7531\u6a19\u7c64\u9023\u7d50\u3002<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"[{\"content\":\"\u7406\u7531\u662f\uff1a\u9996\u5148 \\\\(\\\\overline{AD}\\\\) \u662f 6 \u865f\u62fc\u7247\u7684\u8170\uff0c\u9577\u5ea6\u70ba 1\uff1b\u5176\u659c\u7387\u53ef\u81ea \\\\(\\\\overline{AB}\\\\) \u659c\u7387 \\\\(-45\\\\degree\\\\) \u8d77\u7b97\uff0c\u52a0\u4e0a 10 \u865f\u62fc\u7247\u7684\u9802\u89d2 \\\\(\\\\angle BAC = 30\\\\degree\\\\)\uff0c\u52a0\u4e0a 6 \u865f\u62fc\u7247\u7684\u5e95\u89d2 \\\\(\\\\angle CAD = 45\\\\degree\\\\)\uff0c\u7e3d\u8a08\u662f \\\\(+30\\\\degree\\\\)\u3002\",\"id\":\"2b96d1d8-2e81-4d21-8895-76eb95e32392\"},{\"content\":\"\u9019\u5c31\u662f\u9019\u7bc7\u958b\u982d\u63d0\u5230\u7684\u7b2c\u4e94\u7a2e\u56f0\u96e3\u5ea6\u3002\",\"id\":\"4079b66d-0e5a-40f3-a43f-cc1c6cfb3d7f\"},{\"content\":\"\u9019\u662f\u7b2c\u56db\u7a2e\u56f0\u96e3\u5ea6\u3002\",\"id\":\"e9ab6017-a0c8-498b-8fb3-5d8d3c0209b0\"},{\"content\":\"\u4e03\u516b\u4e5d\u9019\u4e09\u7a2e\u56f0\u96e3\u5ea6\u7684\u8cc7\u6599\u662f\u4f86\u81ea <a href=\\\"https:\/\/www.ptt.cc\/bbs\/puzzle\/M.1312721580.A.B49.html\\\">playertsai \u7684\u6587\u7ae0<\/a>\u3002\",\"id\":\"35237a18-b815-4044-8535-d386fb66f7ca\"}]"},"categories":[3],"tags":[6],"class_list":["post-290","post","type-post","status-publish","format-standard","hentry","category-math","tag-lonpos"],"_links":{"self":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts\/290","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=290"}],"version-history":[{"count":67,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts\/290\/revisions"}],"predecessor-version":[{"id":1745,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts\/290\/revisions\/1745"}],"wp:attachment":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=290"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=290"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}