\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u5186\u306b\u5185\u63a5\u3059\u308b \u25b3ABC \u306e \\(3\\) \u8fba AB, BC, CA \u3092\u305d\u308c\u305e\u308c \\(2 : 3\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092 P, Q, R \u3068\u3059\u308b. \u25b3PQR \u306e\u5916\u5fc3\u304c\u70b9 O \u3068\u4e00\u81f4\u3059\u308b\u3068\u304d, \u25b3ABC \u306f\u3069\u306e\u3088\u3046\u306a\u4e09\u89d2\u5f62\u304b.<\/p>\r\n
\\(\\overrightarrow{a} = \\overrightarrow{\\text{OA}} , \\ \\overrightarrow{b} = \\overrightarrow{\\text{OB}} , \\ \\overrightarrow{c} = \\overrightarrow{\\text{OC}}\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\left| \\overrightarrow{a} \\right| = \\left| \\overrightarrow{b} \\right| = \\left| \\overrightarrow{c} \\right| = k \\quad ( k \\gt 0 )\r\n\\]\r\n\u3068\u304a\u3051\u308b.
\r\nP \u306f AB \u3092 \\(2 : 3\\) \u306b\u5185\u5206\u3059\u308b\u306e\u3067\r\n\\[\r\n\\overrightarrow{\\text{OP}} = \\dfrac{3 \\overrightarrow{a} +2 \\overrightarrow{b}}{5}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{OP}} \\right|^2 & = \\dfrac{9}{25} \\left| \\overrightarrow{a} \\right|^2 +\\dfrac{12}{25} + \\overrightarrow{a} \\cdot \\overrightarrow{b} + \\dfrac{4}{25} \\left| \\overrightarrow{b} \\right|^2 \\\\\r\n& = \\dfrac{12 \\overrightarrow{a} \\cdot \\overrightarrow{b} +13k^2}{25} \\quad ... [1]\r\n\\end{align}\\]\r\n\u540c\u69d8\u306b\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{OQ}} \\right|^2 & = \\dfrac{12 \\overrightarrow{b} \\cdot \\overrightarrow{c} +13k^2}{25} \\quad ... [2] , \\\\\r\n\\left| \\overrightarrow{\\text{OR}} \\right|^2 & = \\dfrac{12 \\overrightarrow{c} \\cdot \\overrightarrow{a} +13k^2}{25} \\quad ... [3]\r\n\\end{align}\\]\r\n\u25b3PQR \u306e\u5916\u5fc3\u304c\u70b9 O \u3068\u306a\u308b\u306e\u3067\r\n\\[\r\n\\left| \\overrightarrow{\\text{OP}} \\right| = \\left| \\overrightarrow{\\text{OQ}} \\right| = \\left| \\overrightarrow{\\text{OR}} \\right|\r\n\\]\r\n\u3067\u3042\u308b\u304b\u3089, [1] \uff5e [3] \u3088\u308a\r\n\\[\r\n\\overrightarrow{a} \\cdot \\overrightarrow{b} = \\overrightarrow{b} \\cdot \\overrightarrow{c} = \\overrightarrow{c} \\cdot \\overrightarrow{a}\r\n\\]\r\n\u3053\u306e\u5f0f\u306e\u5024\u3092 \\(\\ell \\ ( \\ell \\gt 0 )\\) \u3068\u304a\u304f.
\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{AB}} \\right|^2 & = \\left| \\overrightarrow{a} \\right|^2 -2 \\overrightarrow{a} \\cdot \\overrightarrow{b} +\\left| \\overrightarrow{b} \\right|^2 \\\\\r\n& = 2( k -\\ell )\r\n\\end{align}\\]\r\n\u540c\u69d8\u306b\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{BC}} \\right|^2 & = 2( k -\\ell ) , \\\\\r\n\\left| \\overrightarrow{\\text{CA}} \\right|^2 & = 2( k -\\ell )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\left| \\overrightarrow{\\text{AB}} \\right| = \\left| \\overrightarrow{\\text{BC}} \\right| = \\left| \\overrightarrow{\\text{CA}} \\right|\r\n\\]\r\n\u306a\u306e\u3067, \u25b3ABC \u306f\u6b63\u4e09\u89d2\u5f62<\/strong>.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u5186\u306b\u5185\u63a5\u3059\u308b \u25b3ABC \u306e \\(3\\) \u8fba AB, BC, CA \u3092\u305d\u308c\u305e\u308c \\(2 : 3\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092 P, Q, R \u3068\u3059\u308b. \u25b3PQR \u306e\u5916\u5fc3\u304c\u70b9 O \u3068\u4e00\u81f4\u3059\u308b\u3068\u304d, \u25b3ABC \u306f\u3069\u306e … \u7d9a\u304d\u3092\u8aad\u3080