{"id":1255,"date":"2025-07-15T21:08:34","date_gmt":"2025-07-15T12:08:34","guid":{"rendered":"https:\/\/math-friend.com\/?p=1255"},"modified":"2025-08-01T09:49:08","modified_gmt":"2025-08-01T00:49:08","slug":"%e3%80%90%e4%b9%9d%e5%b7%9e%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e5%86%86%e5%91%a8%e4%b8%8a%e3%81%ae%e5%8b%95%e7%82%b9p%e3%81%ab%e9%96%a2%e3%81%99%e3%82%8b%e6%9c%80%e5%a4%a7%e5%80%a4%e5%95%8f","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=1255","title":{"rendered":"\u3010\u4e5d\u5dde\u5927\u5b66\u5165\u8a66\u3011\u5186\u5468\u4e0a\u306e\u52d5\u70b9P\u306b\u95a2\u3059\u308b\u6700\u5927\u5024\u554f\u984c(2025)"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\u534a\u5f84\\(1\\)\u306e\u5186\u5468\u4e0a\u306b2\u70b9\\(\\mathrm{A}\\), \\(\\mathrm{B}\\)\u3092\\(\\mathrm{AB}=\\sqrt{3}\\)\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u3068\u308b. \u3053\u306e\u3068\u304d, \u5186\u5468\u4e0a\u3092\u52d5\u304f\u70b9\\(\\mathrm{P}\\)\u306b\u5bfe\u3057\u3066, \\(\\mathrm{AP}^2+\\mathrm{BP}^2\\)\u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.
(2025 \u4e5d\u5dde\u5927\u5b66\u6587\u7cfb[2])<\/span><\/p>\n\n\n\n

\u9069\u5207\u306b\u5ea7\u6a19\u3092\u8a2d\u5b9a\u3057, \\(\\mathrm{AP}^2+\\mathrm{BP}^2\\)\u3092\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u8868\u305b\u3070\u3059\u3050\u306b\u89e3\u3051\u308b\u554f\u984c\u3067\u3059. \u4eca\u56de\u306f\u56f3\u5f62\u7684\u306a\u5225\u89e3\u3082\u6dfb\u3048\u3066\u307f\u307e\u3057\u305f.

\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n

\n

\\(xy\\)\u5e73\u9762\u306b\u539f\u70b9\\(\\mathrm{O}\\)\u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84\\(1\\)\u306e\u5186\u3092\u63cf\u304d, \u56f3\u306e\u3088\u3046\u306b\u5186\u5468\u4e0a\u306e2\u70b9\\(\\mathrm{A} \\), \\(\\mathrm{B}\\)\u3092, \u5f26\\(\\mathrm{AB}\\)\u304c\\(x\\)\u8ef8\u306b\u5e73\u884c\u3067\\(x\\)\u8ef8\u306e\u4e0b\u65b9\u306b\u4f4d\u7f6e\u3059\u308b\u3088\u3046\u306b\u914d\u7f6e\u3059\u308b. \u5f26\\(\\mathrm{AB}\\)\u306e\u4e2d\u70b9\u3092\\(\\mathrm{M}\\)\u3068\u3059\u308b\u3068, \\(\\displaystyle\\mathrm{MB}=\\frac{\\sqrt{3}}{2}\\), \\(\\mathrm{OB}=1\\)\u3067\u3042\u308a, \\(\\triangle{\\mathrm{OMB}}\\)\u306b\u3066\u4e09\u5e73\u65b9\u306e\u5b9a\u7406\u3092\u7528\u3044\u308b\u3053\u3068\u3067, \\(\\displaystyle \\mathrm{OM}=\\frac{1}{2}\\)\u304c\u308f\u304b\u308a, \\(\\displaystyle \\mathrm{A} \\left(-\\frac{\\sqrt{3}}{2},-\\frac{1}{2}\\right)\\), \\(\\displaystyle \\mathrm{B} \\left(\\frac{\\sqrt{3}}{2},-\\frac{1}{2}\\right)\\)\u3068\u306a\u308b. \u3053\u306e\u3088\u3046\u306b, \\(\\mathrm{A} \\), \\(\\mathrm{B}\\)\u3092\u6c7a\u3081\u3066\u3082\u4e00\u822c\u6027\u3092\u5931\u308f\u306a\u3044. \u307e\u305f, \u5186\u5468\u4e0a\u306e\u70b9\\(\\mathrm{P} \\)\u306b\u5bfe\u3057\u3066, \\(x\\)\u8ef8\u306e\u6b63\u306e\u5411\u304d\u304b\u3089\\(\\mathrm{OP}\\)\u307e\u3067\u53cd\u6642\u8a08\u56de\u308a\u306b\u6e2c\u3063\u305f\u89d2\u306e\u5927\u304d\u3055\u3092\\(\\theta\\)\u3068\u3059\u308b. \\(\\theta\\)\u306e\u7bc4\u56f2\u306f\\(0^\\circ\\leq\\theta<{360}^\\circ\\)\u3067\u3042\u308a, \\(\\mathrm{P}\\)\u306e\u5ea7\u6a19\u306f\\((\\cos{\\theta}, \\sin{\\theta})\\)\u3067\u3042\u308b.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\u4ee5\u4e0a\u306e\u8a2d\u5b9a\u3088\u308a,
$$
\\begin{align}
\\mathrm{AP}^2+\\mathrm{BP}^2&=\\left(\\cos{\\theta}-\\left(-\\frac{\\sqrt{3}}{2}\\right)\\right)^2+\\left(\\sin{\\theta}-\\left(-\\frac{1}{2}\\right)\\right)^2\\\\[1.5ex]
&+\\left(\\cos{\\theta}-\\frac{\\sqrt{3}}{2}\\right)^2+\\left(\\sin{\\theta}-\\frac{1}{2}\\right)^2\\\\[1.5ex]
&=\\cos^2{\\theta}+\\sqrt{3}\\cos{\\theta}+\\frac{3}{4}+\\sin^2{\\theta}+\\sin{\\theta}+\\frac{1}{4}\\\\[1.5ex]
&+\\cos^2{\\theta}-\\sqrt{3}\\cos{\\theta}+\\frac{3}{4}+\\sin^2{\\theta}+\\sin{\\theta}+\\frac{1}{4}\\\\[1.5ex]
&=4+2\\sin{\\theta}
\\end{align}
$$\u3068\u6c42\u307e\u308b.

\u3053\u3053\u3067, \\(\\sin{\\theta}\\)\u306f\\(\\displaystyle \\theta={90}^\\circ\\)\u306e\u3068\u304d, \u6700\u5927\u5024\u3092\u53d6\u308b\u304b\u3089, \\(\\mathrm{AP}^2+\\mathrm{BP}^2\\)\u3082\\(\\displaystyle \\theta={90}^\\circ\\)\u306e\u3068\u304d\u306b\u6700\u5927\u3068\u306a\u308a, \u305d\u306e\u5024\u306f\\(6\\)\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

\u5ea7\u6a19\u5e73\u9762\u4e0a\u306b\u9069\u5207\u306b\u5186, \u70b9\u3092\u914d\u7f6e\u3059\u308b\u3053\u3068\u3067, \u3044\u3068\u3082\u7c21\u5358\u306b\u89e3\u3051\u307e\u3057\u305f. \u3061\u306a\u307f\u306b, \\(\\mathrm{AP}^2+\\mathrm{BP}^2\\)\u304c\u6700\u5927\u306b\u306a\u308b\u306e\u306f, \\(\\mathrm{P}\\)\u304c\u4e0b\u56f3\u306e\u4f4d\u7f6e\u306b\u3042\u308b\u3068\u304d\u3067\u3042\u308a, \u3053\u308c\u3082\u76f4\u611f\u304b\u3089\u306f\u660e\u3089\u304b\u3067\u3059.

\u89e3\u7b54\u304c\u3042\u307e\u308a\u306b\u3082\u77ed\u304b\u3063\u305f\u306e\u3067, \u3082\u3046\u5c11\u3057\u5e7e\u4f55\u306b\u5bc4\u305b\u305f\u5225\u89e3\u3092\u8003\u3048\u307e\u3057\u305f.<\/p>\n\n\n\n

(\u5225\u89e3)
\u4e00\u822c\u306b, \u5e73\u9762\u4e0a\u306b2\u70b9\\(\\mathrm{A}\\), \\(\\mathrm{B}\\)\u304c\u3042\u308a, \u305d\u306e\u4e2d\u70b9\u3092\\(\\mathrm{M}\\)\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u5e73\u9762\u4e0a\u306e\u4efb\u610f\u306e\u70b9\\(\\mathrm{P}\\)\u306b\u5bfe\u3057,
$$
\\mathrm{AP}^2+\\mathrm{BP}^2=2\\mathrm{MP}^2+\\frac{1}{2}\\mathrm{AB}^2\\,\\,\u30fb\u30fb\u30fb\u2460
$$\u304c\u6210\u308a\u7acb\u3064(\u8a3c\u660e\u306f\u5f8c\u8ff0).

\"\"

\u4eca\u56de\u306e\u554f\u984c\u3067\u306f\\(\\mathrm{AB}=\\sqrt{3}\\)\u306f\u56fa\u5b9a\u3060\u304b\u3089, \\(\\mathrm{MP}\\)\u304c\u6700\u5927\u3068\u306a\u308b\u3068\u304d\u306b\\(\\mathrm{AP}^2+\\mathrm{BP}^2\\)\u3082\u307e\u305f\u6700\u5927\u3068\u306a\u308b.

\u554f\u984c\u3067\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u534a\u5f84\\(1\\)\u306e\u5186\u306e\u4e2d\u5fc3\u3092\\(\\mathrm{O}\\)\u3068\u3059\u308b\u3068\u304d, \u4e09\u89d2\u4e0d\u7b49\u5f0f\u304b\u3089,
$$
\\mathrm{MP}\\leq \\mathrm{OM} + \\mathrm{OP}=\\mathrm{OM} + 1
$$\u3068\u306a\u308a, \u6700\u53f3\u8fba\u306f\\(\\mathrm{P}\\)\u306b\u4f9d\u3089\u306a\u3044\u4e00\u5b9a\u5024\u3067\u3042\u308b. \u3053\u306e\u4e0d\u7b49\u5f0f\u306e\u7d71\u5408\u304c\u6210\u7acb\u3059\u308b\u306e\u306f, \\(\\mathrm{M}\\), \\(\\mathrm{O}\\), \\(\\mathrm{P}\\)\u304c\u3053\u306e\u9806\u306b1\u76f4\u7dda\u4e0a\u306b\u4e26\u3076\u3068\u304d\u3067\u3042\u308b.

\u3053\u3053\u3067, \u4ee5\u4e0b\u306e\u56f3\u304b\u3089,

\u4e09\u5e73\u65b9\u306e\u5b9a\u7406\u3088\u308a,
$$
\\mathrm{OM} = \\sqrt{1^2 – \\left( \\frac{\\sqrt{3}}{2} \\right)^2} = \\frac{1}{2}
$$\u3067\u3042\u308a, \u540c\u3058\u304f\u4e09\u5e73\u65b9\u306e\u5b9a\u7406\u304b\u3089,
$$
\\begin{align}
\\mathrm{AP}^2&=\\mathrm{AM}^2+\\mathrm{MP}^2=\\left(\\frac{\\sqrt{3}}{2}\\right)^2+\\left(\\frac{1}{2}+1\\right)^2\\\\[1.5ex]
&=\\frac{3}{4}+\\frac{9}{4}=3
\\end{align}
$$\u3068\u306a\u308a, \u56f3\u306e\u5bfe\u79f0\u6027\u304b\u3089\\(\\mathrm{BP}^2=3\\)\u3067\u3042\u308b.

\u3088\u3063\u3066, \\(\\mathrm{AP}^2+\\mathrm{BP}^2\\)\u306e\u6700\u5927\u5024\u306f\\(3+3=6\\)\u3067\u3042\u308b.<\/p>\n\n\n\n

\u5225\u89e3\u4e2d\u306e\u2460\u306e\u8a3c\u660e\u3092\u4e0e\u3048\u3066\u304a\u304d\u307e\u3059.<\/p>\n\n\n\n

(\u2460\u306e\u8a3c\u660e)
\u4f59\u5f26\u5b9a\u7406\u3088\u308a,
$$
\\begin{align}
\\mathrm{AP}^2=\\mathrm{AM}^2+\\mathrm{MP}^2-2 \\mathrm{AM}\\cdot \\mathrm{MP}\\cos{\\angle{\\mathrm{AMP}}}\\\\[1.5ex]
\\mathrm{BP}^2=\\mathrm{BM}^2+\\mathrm{MP}^2-2 \\mathrm{BM}\\cdot \\mathrm{MP}\\cos{\\angle{\\mathrm{BMP}}}
\\end{align}
$$\u3067\u3042\u308b. \u3053\u3053\u3067,
$$
\\begin{align}
\\cos \\angle \\mathrm{AMP} + \\cos \\angle \\mathrm{BMP}
&= \\cos \\angle \\mathrm{AMP} + \\cos\\left(180^\\circ – \\angle \\mathrm{AMP} \\right) \\\\[1.5ex]
&= \\cos \\angle \\mathrm{AMP} – \\cos \\angle \\mathrm{AMP} \\\\[1.5ex]
&= 0
\\end{align}
$$\u3067\u3042\u308b\u3053\u3068, \u307e\u305f, \\(\\displaystyle \\mathrm{AM}=\\mathrm{BM}=\\frac{\\mathrm{AB}}{2}\\)\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u4e21\u8fba\u3092\u8db3\u305b\u3070,
$$
\\mathrm{AP}^2+\\mathrm{BP}^2=\\mathrm{AM}^2+\\mathrm{BM}^2+2\\mathrm{MP}^2=\\frac{1}{2}\\mathrm{AB}^2+2\\mathrm{MP}^2
$$\u3068\u306a\u308b.<\/p>\n\n\n\n

youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n