{"id":1297,"date":"2015-10-11T21:26:20","date_gmt":"2015-10-11T12:26:20","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1297"},"modified":"2021-09-29T22:41:38","modified_gmt":"2021-09-29T13:41:38","slug":"tbr201502","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr201502\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2015\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u534a\u5f84 \\(1\\) \u306e\u5186\u3092\u5185\u63a5\u5186\u3068\u3059\u308b\u4e09\u89d2\u5f62 ABC \u304c, \u8fba AB \u3068\u8fba AC \u306e\u9577\u3055\u304c\u7b49\u3057\u3044\u4e8c\u7b49\u8fba\u4e09\u89d2\u5f62\u3067\u3042\u308b\u3068\u3059\u308b.\r\n\u8fba BC , CA , AB \u3068\u5185\u63a5\u5186\u306e\u63a5\u70b9\u3092\u305d\u308c\u305e\u308c P , Q , R \u3068\u3059\u308b. \u307e\u305f, \\(\\alpha = \\angle \\text{CAB}\\) , \\(\\beta = \\angle \\text{ABC}\\) \u3068\u3057, \u4e09\u89d2\u5f62 ABC \u306e\u9762\u7a4d\u3092 \\(S\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u7dda\u5206 AQ \u306e\u9577\u3055\u3092 \\(\\alpha\\) \u3092\u7528\u3044\u3066\u8868\u3057, \u7dda\u5206 QC \u306e\u9577\u3055\u3092 \\(\\beta\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(t = \\tan \\dfrac{\\beta}{2}\\) \u3068\u304a\u304f. \u3053\u306e\u3068\u304d, \\(S\\) \u3092 \\(t\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u4e0d\u7b49\u5f0f \\(S \\geqq 3 \\sqrt{3}\\) \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b. \u3055\u3089\u306b, \u7b49\u53f7\u304c\u6210\u7acb\u3059\u308b\u306e\u306f, \u4e09\u89d2\u5f62 ABC \u304c\u6b63\u4e09\u89d2\u5f62\u306e\u3068\u304d\u306b\u9650\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \u5185\u63a5\u5186\u306e\u4e2d\u5fc3\u3092 O \u3068\u304a\u3051\u3070, OP , OQ , OR \u306f\u305d\u308c\u305e\u308c\u9577\u3055 \\(1\\) \u3067, \u5404\u8fba BC , CA , AB \u3068\u5782\u76f4\u3067\u3042\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\text{AQ} = \\underline{\\dfrac{1}{\\tan \\frac{\\alpha}{2}}} , \\quad \\text{CQ} = \\underline{\\dfrac{1}{\\tan \\frac{\\beta}{2}}}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    \u6761\u4ef6\u3088\u308a, \\(0 \\lt t \\lt 1\\) ... [1] .\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{2} \\cdot 1 ( \\text{AB} +\\text{BC} +\\text{CA} ) \\\\\r\n& = \\text{AQ} +2 \\text{CQ}\r\n\\end{align}\\]\r\n\\(\\alpha = \\pi -2 \\beta\\) ... [2] \u3059\u306a\u308f\u3061 \\(\\dfrac{\\alpha}{2} = \\dfrac{\\pi}{2} -\\beta\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\tan \\dfrac{\\alpha}{2} & = \\dfrac{1}{\\tan \\beta} \\\\\r\n& = \\dfrac{1 -\\tan^2 \\frac{\\beta}{2}}{2 \\tan \\frac{\\beta}{2}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, (1)<\/strong> \u306e\u7d50\u679c\u3082\u7528\u3044\u3066\r\n\\[\\begin{align}\r\nS & = \\dfrac{2t}{1 -t^2} +\\dfrac{2}{t} \\\\\r\n& = \\underline{\\dfrac{2}{t -t^3}}\r\n\\end{align}\\]\r\n

    (3)<\/strong><\/p>\r\n

    \\(f(t) = t -t^3\\) \u3068\u304a\u3044\u3066, \\(f(t)\\) \u306e\u53d6\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u8003\u3048\u308b.\r\n\\[\r\nf'(t) = 1 -3t^2 = -3 \\left( t +\\dfrac{1}{\\sqrt{3}} \\right) \\left( t -\\dfrac{1}{\\sqrt{3}} \\right)\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] \u306b\u304a\u3051\u308b \\(f(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} t & (0) & \\cdots & \\dfrac{1}{\\sqrt{3}} & \\cdots & (1) \\\\ \\hline f'(t) & & + & 0 & - & \\\\ \\hline f(t) & (0) & \\nearrow & \\dfrac{2}{3 \\sqrt{3}} & \\searrow & (0) \\end{array}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n0 \\lt f(t) & = \\dfrac{2}{S} \\leqq \\dfrac{2}{3 \\sqrt{3}} \\\\\r\n\\text{\u2234} \\quad & S \\geqq 3 \\sqrt{3}\r\n\\end{align}\\]\r\n\u307e\u305f, \u7b49\u53f7\u6210\u7acb\u306f \\(t = \\tan \\dfrac{\\beta}{2} = \\dfrac{2}{\\sqrt{3}}\\) \u306e\u3068\u304d\u3060\u3051\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{\\beta}{2} & = \\dfrac{\\pi}{6} \\\\\r\n\\text{\u2234} \\quad \\beta & = \\dfrac{\\pi}{3}\r\n\\end{align}\\]\r\n[2] \u3088\u308a\r\n\\[\r\n\\alpha = \\pi -2 \\cdot \\dfrac{\\pi}{3} = \\dfrac{\\pi}{3}\r\n\\]\r\n\u3088\u3063\u3066, \\(S\\) \u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f, \\(\\triangle \\text{ABC}\\) \u304c\u6b63\u4e09\u89d2\u5f62\u306e\u3068\u304d\u306e\u307f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u534a\u5f84 \\(1\\) \u306e\u5186\u3092\u5185\u63a5\u5186\u3068\u3059\u308b\u4e09\u89d2\u5f62 ABC \u304c, \u8fba AB \u3068\u8fba AC \u306e\u9577\u3055\u304c\u7b49\u3057\u3044\u4e8c\u7b49\u8fba\u4e09\u89d2\u5f62\u3067\u3042\u308b\u3068\u3059\u308b. \u8fba BC , CA , AB \u3068\u5185\u63a5\u5186\u306e\u63a5\u70b9\u3092\u305d\u308c\u305e\u308c P , Q , R \u3068\u3059\u308b. \u307e\u305f, \\( … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[132],"tags":[144,137],"class_list":["post-1297","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2015","tag-tsukuba_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1297","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=1297"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1297\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1297"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1297"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1297"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}