{"id":767,"date":"2013-05-05T09:17:50","date_gmt":"2013-05-05T00:17:50","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=767"},"modified":"2021-03-10T16:21:02","modified_gmt":"2021-03-10T07:21:02","slug":"tkr201304","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201304\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2013\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\u25b3ABC \u306b\u304a\u3044\u3066, \\(\\angle \\text{BAC} = 90^{\\circ}\\) , \\(\\left| \\overrightarrow{\\text{AB}} \\right| = 1\\) , \\(\\left| \\overrightarrow{\\text{AC}} \\right| = \\sqrt{3}\\) \u3068\u3059\u308b. \u25b3ABC \u306e\u5185\u90e8\u306e\u70b9 P \u304c\r\n\\[\r\n\\dfrac{\\overrightarrow{\\text{PA}}}{\\left| \\overrightarrow{\\text{PA}} \\right|} +\\dfrac{\\overrightarrow{\\text{PB}}}{\\left| \\overrightarrow{\\text{PB}} \\right|} +\\dfrac{\\overrightarrow{\\text{PC}}}{\\left| \\overrightarrow{\\text{PC}} \\right|} = \\overrightarrow{0}\r\n\\]\r\n\u3092\u6e80\u305f\u3059\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\angle \\text{APB}\\) , \\(\\angle \\text{APC}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\left| \\overrightarrow{\\text{PA}} \\right|\\) , \\(\\left| \\overrightarrow{\\text{PB}} \\right|\\) , \\(\\left| \\overrightarrow{\\text{PC}} \\right|\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

    \\[\r\n\\dfrac{\\overrightarrow{\\text{PA}}}{\\left| \\overrightarrow{\\text{PA}} \\right|} +\\dfrac{\\overrightarrow{\\text{PB}}}{\\left| \\overrightarrow{\\text{PB}} \\right|} +\\dfrac{\\overrightarrow{\\text{PC}}}{\\left| \\overrightarrow{\\text{PC}} \\right|} = \\overrightarrow{0} \\quad ...\\text{[A]}\r\n\\]\r\n[A] \u306e\u4e21\u8fba\u306b\u3064\u3044\u3066, \\(\\overrightarrow{\\text{PA}}\\) \u3068\u306e\u5185\u7a4d\u3092\u8003\u3048\u308b\u3068\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{PA}} \\cdot \\left( \\dfrac{\\overrightarrow{\\text{PA}}}{\\left| \\overrightarrow{\\text{PA}} \\right|} +\\dfrac{\\overrightarrow{\\text{PB}}}{\\left| \\overrightarrow{\\text{PB}} \\right|} +\\dfrac{\\overrightarrow{\\text{PC}}}{\\left| \\overrightarrow{\\text{PC}} \\right|} \\right) = \\overrightarrow{\\text{PA}} & \\cdot \\overrightarrow{0} \\\\\r\n\\left| \\overrightarrow{\\text{PA}} \\right| + \\left| \\overrightarrow{\\text{PA}} \\right| \\cos \\angle \\text{APB} + \\left| \\overrightarrow{\\text{PA}} \\right| \\cos \\angle \\text{APC} & = 0\r\n\\end{align}\\]\r\n\\(\\left| \\overrightarrow{\\text{PA}} \\right| \\neq 0\\) \u306a\u306e\u3067\r\n\\[\r\n\\cos \\angle \\text{APB} + \\cos \\angle \\text{APC} = -1 \\quad ... [1]\r\n\\]\r\n\u540c\u69d8\u306b, [A] \u306e\u4e21\u8fba\u306b\u3064\u3044\u3066, \\(\\overrightarrow{\\text{PB}} , \\overrightarrow{\\text{PC}}\\) \u3068\u306e\u5185\u7a4d\u3092\u8003\u3048\u308c\u3070\r\n\\[\\begin{align}\r\n\\cos \\angle \\text{BPC} + \\cos \\angle \\text{APC} = -1 \\quad ... [2] \\\\\r\n\\cos \\angle \\text{APB} + \\cos \\angle \\text{BPC} = -1 \\quad ... [3]\r\n\\end{align}\\]\r\n[1] \uff5e [3] \u3092\u3068\u304f\u3068\r\n\\[\r\n\\cos \\angle \\text{APB} = \\cos \\angle \\text{APC} = \\cos \\angle \\text{APB} \uff1d-\\dfrac{1}{2}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\text{APB} = \\angle \\text{APC} = \\angle \\text{APB} = \\underline{120^{\\circ}}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n\"tokyo_r_2013_04_01\"\r\n

    \\(x = \\left| \\overrightarrow{\\text{PA}} \\right|\\) , \\(y = \\left| \\overrightarrow{\\text{PB}} \\right|\\) , \\(z = \\left| \\overrightarrow{\\text{PC}} \\right|\\) \u3068\u304a\u304f.
    \r\n\u307e\u305f, \\(\\alpha = \\angle \\text{PAB}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\angle \\text{PAC} & = 90^{\\circ} -\\alpha \\\\\r\n\\angle \\text{PBA} & = 180^{\\circ} -120^{\\circ} -\\alpha = 60^{\\circ} -\\alpha\r\n\\end{align}\\]\r\n\u307e\u305f, \u6761\u4ef6\u3088\u308a \\(\\angle \\text{ABC} = 60^{\\circ}\\) .\r\n\\[\r\n\\angle \\text{PBC} = 60^{\\circ} -( 60^{\\circ} -\\alpha ) = \\alpha\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\angle \\text{PAB} = \\angle \\text{PBC}\r\n\\]\r\n\u3053\u308c\u3068 (1)<\/strong> \u306e\u7d50\u679c\u304b\u3089, \\(2\\) \u89d2\u304c\u7b49\u3057\u3044\u306e\u3067\r\n\\[\r\n\\triangle \\text{ABP} \\sim \\triangle \\text{BCP}\r\n\\]\r\n\u3055\u3089\u306b \\(\\text{AB} = 1\\) , \\(\\text{BC} = 2\\) \u306a\u306e\u3067, \u76f8\u4f3c\u6bd4\u306f \\(1 : 2\\) \u3060\u304b\u3089\r\n\\[\\begin{align}\r\nx : y & = y : z = 1 : 2 \\\\\r\n\\text{\u2234} \\quad y & = 2x , \\quad z = 4x \\quad ... [4]\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(\\triangle \\text{ABP}\\) \u306b\u3064\u3044\u3066, \u4f59\u5f26\u5b9a\u7406\u304b\u3089\r\n\\[\\begin{align}\r\nx^2 +(2x)^2 -2 x & \\cdot 2x \\cdot \\cos 120^{\\circ} = 1 \\\\\r\n7x^2 & = 1 \\\\\r\n\\text{\u2234} \\quad x & = \\dfrac{\\sqrt{7}}{7}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, [4] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\r\n\\left| \\overrightarrow{\\text{PA}} \\right| = \\underline{\\dfrac{\\sqrt{7}}{7}} , \\quad \\left| \\overrightarrow{\\text{PB}} \\right| = \\underline{\\dfrac{2 \\sqrt{7}}{7}} , \\quad \\left| \\overrightarrow{\\text{PC}} \\right| = \\underline{\\dfrac{4 \\sqrt{7}}{7}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u25b3ABC \u306b\u304a\u3044\u3066, \\(\\angle \\text{BAC} = 90^{\\circ}\\) , \\(\\left| \\overrightarrow{\\text{AB}} \\right| = 1\\) , \\(\\left| \\ … \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":[86],"tags":[139,111],"class_list":["post-767","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2013","tag-tokyo_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/767","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=767"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/767\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=767"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=767"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=767"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}