{"id":72,"date":"2011-11-26T22:13:26","date_gmt":"2011-11-26T13:13:26","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=72"},"modified":"2021-11-04T09:30:34","modified_gmt":"2021-11-04T00:30:34","slug":"htb201104","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb201104\/","title":{"rendered":"\u4e00\u6a4b\u59272011\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(a , b , c\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3059\u308b.\r\n\u7a7a\u9593\u5185\u306b \\(3\\) \u70b9 A \\(( a , 0 , 0 )\\) , B \\(( 0 , b , 0 )\\) , C \\(( 0 , 0 , c )\\) \u304c\u3042\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u8fba AB \u3092\u5e95\u8fba\u3068\u3059\u308b\u3068\u304d, \u25b3ABC \u306e\u9ad8\u3055\u3092 \\(a , b , c\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u25b3ABC , \u25b3OAB , \u25b3OBC , \u25b3OCA \u306e\u9762\u7a4d\u3092\u305d\u308c\u305e\u308c \\(S , S _ 1 , S _ 2 , S _ 3\\) \u3068\u3059\u308b. \u305f\u3060\u3057, O \u306f\u539f\u70b9\u3067\u3042\u308b. \u3053\u306e\u3068\u304d, \u4e0d\u7b49\u5f0f \\(\\sqrt{3} S \\geqq S _ 1 +S _ 2 +S _ 3\\) \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000(2)<\/strong> \u306e\u4e0d\u7b49\u5f0f\u306b\u304a\u3044\u3066\u7b49\u53f7\u304c\u6210\u308a\u7acb\u3064\u305f\u3081\u306e\u6761\u4ef6\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

    \\(\\overrightarrow{\\text{CA}} = \\left( \\begin{array}{c} -a \\\\ 0 \\\\ c \\end{array} \\right)\\) , \\(\\overrightarrow{\\text{CB}} = \\left( \\begin{array}{c} 0 \\\\ -b \\\\ c \\end{array} \\right)\\) \u306a\u306e\u3067\r\n\\[\r\n\\overrightarrow{\\text{CA}} \\cdot \\overrightarrow{\\text{CA}} = c^2\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\triangle \\text{ABC} & = \\dfrac{1}{2} \\sqrt{ \\left| \\overrightarrow{\\text{CA}} \\right|^2 \\left| \\overrightarrow{\\text{CB}} \\right|^2 -\\left( \\overrightarrow{\\text{CA}} \\cdot \\overrightarrow{\\text{CB}} \\right)^2} \\\\\r\n& = \\dfrac{1}{2} \\sqrt{(a^2+c^2) +(b^2+c^2) -c^4} \\\\\r\n& = \\dfrac{1}{2} \\sqrt{a^2 b^2 +b^2 c^2 +c^2 a^2} \\quad ... [1]\r\n\\end{align}\\]\r\n\u307e\u305f, \u6c42\u3081\u308b\u9ad8\u3055\u3092 \\(h\\) \u3068\u304a\u3051\u3070, \\(\\triangle \\text{ABC} = \\dfrac{1}{2} h \\sqrt{a^2+b^2}\\) \u306a\u306e\u3067\r\n\\[\r\nh = \\underline{\\sqrt{\\dfrac{a^2 b^2 +b^2 c^2 +c^2 a^2}{a^2+b^2}}}\r\n\\]\r\n

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

    \\[\\begin{align}\r\nS _ 1 +S _ 2 +S _ 3 & = \\dfrac{1}{2} (ab+bc+ca) = \\dfrac{1}{2} \\left( \\begin{array}{c} 1 \\\\ 1 \\\\ 1 \\end{array} \\right) \\cdot \\left( \\begin{array}{c} ab \\\\ bc \\\\ ca \\end{array} \\right) \\\\\r\n& \\leqq \\dfrac{1}{2} \\sqrt{1^2+1^2+1^2} \\sqrt{a^2 b^2 +b^2 c^2 +c^2 a^2} \\quad ( \\ \\text{\u2235} \\ \\text{\u4e09\u89d2\u4e0d\u7b49\u5f0f} \\ ) \\\\\r\n& = \\sqrt{3} S \\quad ( \\ \\text{\u2235} \\ [1] \\ )\r\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\r\n\\sqrt{3} S \\geqq S _ 1 +S _ 2 +S _ 3\r\n\\]\r\n

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

    (2)<\/strong> \u306e\u7d50\u679c\u306b\u304a\u3044\u3066, \u7b49\u5f0f\u304c\u6210\u7acb\u3059\u308b\u306e\u306f\r\n\\[\r\n\\left( \\begin{array}{c} 1 \\\\ 1 \\\\ 1 \\end{array} \\right) \/ \\! \/ \\left( \\begin{array}{c} ab \\\\ bc \\\\ ca \\end{array} \\right)\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(ab=bc=ca\\) \u3059\u306a\u308f\u3061 \\(\\underline{a=b=c}\\) \u306e\u3068\u304d.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(a , b , c\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3059\u308b. \u7a7a\u9593\u5185\u306b \\(3\\) \u70b9 A \\(( a , 0 , 0 )\\) , B \\(( 0 , b , 0 )\\) , C \\(( 0 , 0 , c )\\) \u304c\u3042\u308b. (1)\u3000 … \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":[47],"tags":[146,13],"class_list":["post-72","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2011","tag-hitotsubashi","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/72","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=72"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/72\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=72"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=72"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=72"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}