{"id":1902,"date":"2021-09-22T21:17:17","date_gmt":"2021-09-22T12:17:17","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1902"},"modified":"2021-09-22T21:17:17","modified_gmt":"2021-09-22T12:17:17","slug":"tbr201603","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr201603\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2016\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\u56db\u9762\u4f53 OABC \u306b\u304a\u3044\u3066, \\(\\overrightarrow{\\text{OA}} = \\overrightarrow{a}\\) , \\(\\overrightarrow{\\text{OB}} = \\overrightarrow{b}\\) , \\(\\overrightarrow{\\text{OC}} = \\overrightarrow{c}\\) \u3068\u304a\u304f. \u3053\u306e\u3068\u304d\u7b49\u5f0f\r\n\\[\r\n\\overrightarrow{a} \\cdot \\overrightarrow{b} = \\overrightarrow{b} \\cdot \\overrightarrow{c} = \\overrightarrow{c} \\cdot \\overrightarrow{a} = 1\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3068\u3059\u308b. \\(t\\) \u306f\u5b9f\u6570\u306e\u5b9a\u6570\u3067, \\(0 \\lt t \\lt 1\\) \u3092\u6e80\u305f\u3059\u3068\u3059\u308b. \u7dda\u5206 OA \u3092 \\(t : 1-t\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092 P \u3068\u3057, \u7dda\u5206 BC \u3092 \\(t : 1-t\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092 Q \u3068\u3059\u308b. \u307e\u305f, \u7dda\u5206 PQ \u306e\u4e2d\u70b9\u3092 M \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\overrightarrow{\\text{OM}}\\) \u3092 \\(\\overrightarrow{a} , \\overrightarrow{b} , \\overrightarrow{c}\\) \u3068 \\(t\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u7dda\u5206 OM \u3068\u7dda\u5206 BM \u306e\u9577\u3055\u304c\u7b49\u3057\u3044\u3068\u304d, \u7dda\u5206 OB \u306e\u9577\u3055\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(4\\) \u70b9 O , A , B , C \u304c\u70b9 M \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u540c\u4e00\u7403\u9762\u4e0a\u306b\u3042\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(\\triangle \\text{OAB}\\) \u3068 \\(\\triangle \\text{OCB}\\) \u306f\u5408\u540c\u3067\u3042\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

    \\(\\overrightarrow{\\text{OP}} = t \\overrightarrow{a}\\) , \\(\\overrightarrow{\\text{OQ}} = ( 1-t ) \\overrightarrow{b} +t \\overrightarrow{c}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OM}} & = \\dfrac{1}{2} \\left( \\overrightarrow{\\text{OP}} +\\overrightarrow{\\text{OQ}} \\right) \\\\\r\n& = \\underline{\\dfrac{t}{2} \\overrightarrow{a} +\\dfrac{1-t}{2} \\overrightarrow{b} +\\dfrac{t}{2} \\overrightarrow{c}} \\ .\r\n\\end{align}\\]\r\n

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

    \\(\\text{OM} = \\text{BM}\\) \u3088\u308a\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{OM}} \\right|^2 -\\left| \\overrightarrow{\\text{BM}} \\right|^2 & = \\left( \\overrightarrow{\\text{OM}} +\\overrightarrow{\\text{BM}} \\right) \\cdot \\left( \\overrightarrow{\\text{OM}} +\\overrightarrow{\\text{BM}} \\right) \\\\\r\n& = \\left( 2 \\overrightarrow{\\text{OM}} -\\overrightarrow{\\text{OB}} \\right) \\cdot \\overrightarrow{\\text{OB}} \\\\\r\n& = t \\left( \\overrightarrow{a} -\\overrightarrow{b} +\\overrightarrow{c} \\right) \\cdot \\overrightarrow{b} \\\\\r\n& = t \\left( 2 -\\left| \\overrightarrow{b} \\right|^2 \\right) = 0 \\ .\r\n\\end{align}\\]\r\n\\(t \\neq 0\\) \u306a\u306e\u3067\r\n\\[\r\n\\left| \\overrightarrow{b} \\right| = \\text{OB} = \\underline{\\sqrt{2}} \\ .\r\n\\]\r\n

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

    \\(\\text{OM} = \\text{AM}\\) \u3088\u308a\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{OM}} \\right|^2 -\\left| \\overrightarrow{\\text{AM}} \\right|^2 & = \\left( \\overrightarrow{\\text{OM}} +\\overrightarrow{\\text{AM}} \\right) \\cdot \\left( \\overrightarrow{\\text{OM}} +\\overrightarrow{\\text{AM}} \\right) \\\\\r\n& = \\left( 2 \\overrightarrow{\\text{OM}} -\\overrightarrow{\\text{OA}} \\right) \\cdot \\overrightarrow{\\text{OA}} \\\\\r\n& = \\left\\{ -(1-t) \\overrightarrow{a} +(1-t) \\overrightarrow{b} +t \\overrightarrow{c} \\right\\} \\cdot \\overrightarrow{a} \\\\\r\n& = 1 -(1-t) \\left| \\overrightarrow{a} \\right|^2 = 0 \\ .\r\n\\end{align}\\]\r\n\\(t \\neq 0\\) \u306a\u306e\u3067\r\n\\[\r\n\\left| \\overrightarrow{a} \\right| = \\text{OA} = \\dfrac{1}{\\sqrt{1-t}} \\ .\r\n\\]\r\n\\(\\text{OM} = \\text{CM}\\) \u3088\u308a, \u540c\u69d8\u306b\u3059\u308c\u3070\r\n\\[\r\n\\left| \\overrightarrow{c} \\right| = \\text{OC} = \\dfrac{1}{\\sqrt{1-t}} \\ .\r\n\\]\r\n\u3086\u3048\u306b, \\(\\text{OA} = \\text{OC}\\) ... [1] .
    \r\n\u6761\u4ef6\u3088\u308a\r\n\\[\\begin{gather}\r\n\\overrightarrow{a} \\cdot \\overrightarrow{b} = \\dfrac{\\sqrt{2}}{\\sqrt{1-t}} \\cos \\angle \\text{AOB} = 1 \\\\\r\n\\text{\u2234} \\quad \\cos \\angle \\text{AOB} = \\dfrac{\\sqrt{1-t}}{\\sqrt{2}} \\ .\r\n\\end{gather}\\]\r\n\\(\\overrightarrow{a} \\cdot \\overrightarrow{c} = 1\\) \u3088\u308a, \u540c\u69d8\u306b\u3059\u308c\u3070\r\n\\[\r\n\\cos \\angle \\text{COB} = \\dfrac{\\sqrt{1-t}}{\\sqrt{2}} \\ .\r\n\\]\r\n\u3086\u3048\u306b, \\(\\angle \\text{AOB} = \\angle \\text{COB}\\) ... [2] .
    \r\n\u25b3OAB \u3068 \u25b3OCB \u306f, OB \u3092\u5171\u6709\u3057, [1] [2] \u3088\u308a, \\(2\\) \u8fba\u3068\u305d\u306e\u9593\u306e\u89d2\u304c\u305d\u308c\u305e\u308c\u7b49\u3057\u3044\u306e\u3067\r\n\\[\r\n\\triangle \\text{OAB} \\equiv \\triangle \\text{OCB} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u56db\u9762\u4f53 OABC \u306b\u304a\u3044\u3066, \\(\\overrightarrow{\\text{OA}} = \\overrightarrow{a}\\) , \\(\\overrightarrow{\\text{OB}} = \\overright … \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":[158],"tags":[144,162],"class_list":["post-1902","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2016","tag-tsukuba_r","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1902","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=1902"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1902\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1902"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1902"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1902"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}