{"id":428,"date":"2012-06-14T23:44:49","date_gmt":"2012-06-14T14:44:49","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=428"},"modified":"2021-10-20T14:56:27","modified_gmt":"2021-10-20T05:56:27","slug":"ykr201205","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201205\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2012\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u92ed\u89d2\u4e09\u89d2\u5f62 ABC \u306e \\(\\angle \\text{A} , \\angle \\text{B} , \\angle \\text{C}\\) \u306e\u5927\u304d\u3055\u3092\u305d\u308c\u305e\u308c \\(\\alpha , \\beta , \\gamma\\) \u3067\u8868\u3059.\r\n\u70b9 D , E , F \u306f\u305d\u308c\u305e\u308c\u8fba CA , AB , BC \u4e0a\u306b\u3042\u308a, \\(\\text{DE} \\perp \\text{AB}\\) , \\(\\text{EF} \\perp \\text{BC}\\) , \\(\\text{FD} \\perp \\text{CA}\\) \u3092\u6e80\u305f\u3059. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u25b3ABC \u3068 \u25b3DEF \u306f\u76f8\u4f3c\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\dfrac{\\text{BC}}{\\text{EF}} = \\dfrac{1}{\\tan \\alpha} +\\dfrac{1}{\\tan \\beta} +\\dfrac{1}{\\tan \\gamma}\\) \u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(\\alpha\\) \u304c\u4e00\u5b9a\u306e\u3068\u304d, \\(\\dfrac{\\text{BC}}{\\text{EF}}\\) \u3092\u6700\u5c0f\u306b\u3059\u308b\u3088\u3046\u306a \\(\\beta , \\gamma\\) \u3092 \\(\\alpha\\) \u3067\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n\"yokokoku_r_2012_05_01\"\r\n

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

    \\[\r\n\\angle \\text{FDE} = \\dfrac{\\pi}{2} -\\angle \\text{ADE} =\\dfrac{\\pi}{2} -\\left( \\dfrac{\\pi}{2} -\\alpha \\right) =\\alpha\r\n\\]\r\n\u540c\u69d8\u306b\u3059\u308c\u3070\r\n\\[\r\n\\angle \\text{DEF} = \\dfrac{\\pi}{2} -\\angle \\text{BEF} = \\dfrac{\\pi}{2} -\\left( \\dfrac{\\pi}{2} -\\beta \\right) = \\beta\r\n\\]\r\n\u3088\u3063\u3066, \\(2\\) \u89d2\u304c\u305d\u308c\u305e\u308c\u7b49\u3057\u3044\u306e\u3067\r\n\\[\r\n\\triangle \\text{ABC} \\sim \\triangle \\text{DEF}\r\n\\]\r\n

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

    \\(\\text{BC} = a\\) , \\(\\text{AC} = b\\) \u3068\u304a\u304f.
    \r\n\u6b63\u5f26\u5b9a\u7406\u3088\u308a\r\n\\[\\begin{align}\r\n\\dfrac{a}{\\sin \\alpha} & = \\dfrac{b}{\\sin \\beta} \\quad ... [1] \\\\\r\n\\text{\u2234} \\quad b & = \\dfrac{a \\sin \\beta}{\\sin \\alpha}\r\n\\end{align}\\]\r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(\\dfrac{\\text{BC}}{\\text{EF}} = \\dfrac{\\text{CA}}{\\text{FD}} = k\\) \u3068\u304a\u3051\u308b\u306e\u3067\r\n\\[\\begin{align}\r\n\\text{BF} & = \\dfrac{\\text{EF}}{\\tan \\beta} = \\dfrac{a}{k \\tan \\beta} \\quad , \\\\\r\n\\text{FC} & = \\dfrac{\\text{FD}}{\\sin \\gamma} = \\dfrac{a \\sin \\beta}{k \\sin \\alpha \\sin \\gamma} \\quad ( \\ \\text{\u2235} \\ [1] \\ )\r\n\\end{align}\\]\r\n\\(\\text{BC} = \\text{BF} +\\text{FC}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\na & = \\dfrac{a}{k \\tan \\beta} +\\dfrac{a \\sin \\beta}{k \\sin \\alpha \\sin \\gamma} \\\\\r\n\\text{\u2234} \\quad k & = \\dfrac{1}{\\tan \\beta} +\\dfrac{\\sin \\beta}{\\sin \\alpha \\sin \\gamma}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\n\\sin \\beta & = \\sin \\left\\{ \\pi -( \\alpha +\\gamma ) \\right\\} =\\sin ( \\alpha +\\gamma ) \\\\\r\n& = \\sin \\alpha \\cos \\gamma +\\cos \\alpha \\sin \\gamma\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nk & = \\dfrac{1}{\\tan \\beta} +\\dfrac{\\sin \\alpha \\cos \\gamma +\\cos \\alpha \\sin \\gamma}{\\sin \\alpha \\sin \\gamma} \\\\\r\n& = \\dfrac{1}{\\tan \\beta} +\\dfrac{\\cos \\alpha}{\\sin \\alpha} +\\dfrac{\\cos \\gamma}{\\sin \\gamma} \\\\\r\n& = \\dfrac{1}{\\tan \\alpha} +\\dfrac{1}{\\tan \\beta} +\\dfrac{1}{\\tan \\gamma}\r\n\\end{align}\\]\r\n

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

    (2)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\nk = \\dfrac{1}{\\tan \\alpha} +\\dfrac{\\sin \\alpha}{\\sin \\beta \\sin \\gamma}\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\n\\sin \\beta \\sin \\gamma & = -\\dfrac{1}{2} \\left\\{ \\cos ( \\beta +\\gamma ) -\\cos ( \\beta -\\gamma ) \\right\\} \\\\\r\n& = -\\dfrac{1}{2} \\left\\{ \\cos ( \\pi -\\alpha ) -\\cos ( \\beta -\\gamma ) \\right\\} \\\\\r\n& = \\dfrac{1}{2} \\left\\{ \\cos \\alpha +\\cos ( \\beta -\\gamma ) \\right\\}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nk =\\dfrac{1}{\\tan \\alpha} +\\dfrac{2 \\sin \\alpha}{\\cos \\alpha +\\cos ( \\beta -\\gamma )}\r\n\\]\r\n\\(\\alpha\\) \u306f\u5b9a\u6570\u306a\u306e\u3067, \\(k\\) \u304c\u6700\u5c0f\u306b\u306a\u308b\u306e\u306f, \\(\\cos ( \\beta -\\gamma )\\) \u304c\u6700\u5927\u306b\u306a\u308b\u3068\u304d\u3067\r\n\\[\r\n\\beta = \\gamma = \\underline{\\dfrac{\\pi -\\alpha}{2}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u92ed\u89d2\u4e09\u89d2\u5f62 ABC \u306e \\(\\angle \\text{A} , \\angle \\text{B} , \\angle \\text{C}\\) \u306e\u5927\u304d\u3055\u3092\u305d\u308c\u305e\u308c \\(\\alpha , \\beta , \\gamma\\) \u3067\u8868\u3059. … \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":[66],"tags":[68,9],"class_list":["post-428","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2012","tag-68","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/428","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=428"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/428\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=428"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=428"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=428"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}