{"id":565,"date":"2025-07-09T19:17:16","date_gmt":"2025-07-09T10:17:16","guid":{"rendered":"https:\/\/math-friend.com\/?p=565"},"modified":"2025-08-01T09:34:32","modified_gmt":"2025-08-01T00:34:32","slug":"%e3%80%90%e4%bf%a1%e5%b7%9e%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e6%ad%a3%e5%9b%9b%e9%9d%a2%e4%bd%93%e3%81%ab%e5%a4%96%e6%8e%a5%e3%81%99%e3%82%8b%e7%90%83%e3%81%ae%e4%b8%ad%e5%bf%832025","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=565","title":{"rendered":"\u3010\u4fe1\u5dde\u5927\u5b66\u5165\u8a66\u3011\u6b63\u56db\u9762\u4f53\u306b\u5916\u63a5\u3059\u308b\u7403\u306e\u4e2d\u5fc3(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

\u56db\u9762\u4f53\\(\\mathrm{ABCD}\\)\u306b\u3064\u3044\u3066, \\(\\triangle{\\mathrm{BCD}}\\)\u306e\u91cd\u5fc3\u3092\\(\\mathrm{G}\\), \u7dda\u5206\\(\\mathrm{AG}\\)\u3092\\(3:1\\)\u306b\u5185\u5206\u3059\u308b\u70b9\u3092\\(\\mathrm{I}\\)\u3068\u3059\u308b\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.
(1) \\(\\overrightarrow{\\mathrm{AI}}\\)\u3092\\(\\overrightarrow{\\mathrm{AB}}\\), \\(\\overrightarrow{\\mathrm{AC}}\\), \\(\\overrightarrow{\\mathrm{AD}}\\)\u3092\u7528\u3044\u3066\u8868\u305b.
(2) \u56db\u9762\u4f53\\(\\mathrm{ABCD}\\)\u304c\u6b63\u56db\u9762\u4f53\u3067\u3042\u308c\u3070, \\(\\mathrm{AI}=\\mathrm{BI}=\\mathrm{CI}=\\mathrm{DI}\\)\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b.
(2025 \u4fe1\u5dde\u5927\u5b66\u6587\u7cfb[2])<\/span><\/p>\n\n\n\n

\u3053\u3061\u3089\u30d9\u30af\u30c8\u30eb\u306e\u57fa\u672c\u7684\u306a\u554f\u984c\u3067\u3059. \u56db\u9762\u4f53\\(\\mathrm{ABCD}\\)\u304c\u6b63\u56db\u9762\u4f53\u306e\u3068\u304d, \\(\\mathrm{AI}=\\mathrm{BI}=\\mathrm{CI}=\\mathrm{EI}\\)\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u554f\u984c\u3067\u3059\u304c, \u3053\u306e\u7d50\u679c\u304b\u3089\\(\\mathrm{I}\\)\u304c4\u70b9\\(\\mathrm{A}\\), \\(\\mathrm{B}\\), \\(\\mathrm{C}\\), \\(\\mathrm{D}\\)\u3092\u901a\u308b\u7403(\u3064\u307e\u308a\u6b63\u56db\u9762\u4f53\\(\\mathrm{ABCD}\\)\u306e\u5916\u63a5\u7403)\u306e\u4e2d\u5fc3\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059.

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

\n

(1) \\(\\mathrm{G}\\)\u306f\\(\\triangle{\\mathrm{BCD}} \\)\u306e\u91cd\u5fc3\u3088\u308a,
$$
\\overrightarrow{\\mathrm{AG}}=\\frac{1}{3}\\left(\\overrightarrow{\\mathrm{AB}}+\\overrightarrow{\\mathrm{AC}}+\\overrightarrow{\\mathrm{AD}}\\right)
$$
\u304c\u6210\u308a\u7acb\u3064. \u307e\u305f, \\(\\mathrm{I}\\)\u306f\\(\\mathrm{AG}\\)\u3092\\(3:1\\)\u306b\u5185\u5206\u3059\u308b\u306e\u3067,
$$
\\overrightarrow{\\mathrm{AI}}=\\frac{3}{4}\\overrightarrow{\\mathrm{AG}}=\\frac{1}{4}\\left(\\overrightarrow{\\mathrm{AB}}+\\overrightarrow{\\mathrm{AC}}+\\overrightarrow{\\mathrm{AD}}\\right)
$$
\u3068\u306a\u308b.

(2) \u56db\u9762\u4f53\\(\\mathrm{ABCD}\\)\u304c\u6b63\u56db\u9762\u4f53\u3067\u3042\u308c\u3070, \\(\\triangle{\\mathrm{ABC}}\\), \\(\\triangle{\\mathrm{ACD}}\\), \\(\\triangle{\\mathrm{ADB}}\\)\u306f\u3044\u305a\u308c\u3082\u6b63\u4e09\u89d2\u5f62\u3068\u306a\u308a, \\(\\overrightarrow{\\mathrm{AB}}\\), \\(\\overrightarrow{\\mathrm{AC}}\\), \\(\\overrightarrow{\\mathrm{AD}}\\)\u306e\u3044\u305a\u308c\u306e2\u3064\u306e\u306a\u3059\u89d2\u3082\\({60}^\\circ\\)\u3067\u3042\u308b. \u307e\u305f, \u3053\u308c\u30893\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u9577\u3055\u3082\u7b49\u3057\u304f\u306a\u308a, \u305d\u308c\u3092\\(a\\)\u3068\u304a\u304f\u3068,
$$
\\begin{align}
\\overrightarrow{\\mathrm{AB}}\\cdot \\overrightarrow{\\mathrm{AC}} = \\left|\\overrightarrow{\\mathrm{AB}}\\right|\\left|\\overrightarrow{\\mathrm{AC}}\\right|\\cos{{60}^\\circ}=\\frac{a^2}{2}
\\end{align}
$$
\u3067\u3042\u308b.

\u3088\u3063\u3066,
$$
\\begin{align}
\\left|\\overrightarrow{\\mathrm{AI}} \\right|^2&=\\left|\\frac{1}{4}\\left(\\overrightarrow{\\mathrm{AB}}+\\overrightarrow{\\mathrm{AC}}+\\overrightarrow{\\mathrm{AD}}\\right)\\right|^2\\\\[1.5ex]
&=\\left(\\frac{1}{4}\\left(\\overrightarrow{\\mathrm{AB}}+\\overrightarrow{\\mathrm{AC}}+\\overrightarrow{\\mathrm{AD}}\\right)\\right)\\cdot \\left(\\frac{1}{4}\\left(\\overrightarrow{\\mathrm{AB}}+\\overrightarrow{\\mathrm{AC}}+\\overrightarrow{\\mathrm{AD}}\\right)\\right)\\\\[1.5ex]
&=\\frac{1}{16}\\left(\\left|\\overrightarrow{\\mathrm{AB}} \\right|^2+\\left|\\overrightarrow{\\mathrm{AC}} \\right|^2+\\left|\\overrightarrow{\\mathrm{AD}} \\right|^2+2 \\overrightarrow{\\mathrm{AB}}\\cdot \\overrightarrow{\\mathrm{AC}}+2 \\overrightarrow{\\mathrm{AC}}\\cdot \\overrightarrow{\\mathrm{AD}}+2 \\overrightarrow{\\mathrm{AD}}\\cdot \\overrightarrow{\\mathrm{AB}} \\right)\\\\[1.5ex]
&=\\frac{1}{16}\\left(a^2+a^2+a^2+2\\cdot\\frac{a^2}{2}+2\\cdot\\frac{a^2}{2}+2\\cdot\\frac{a^2}{2}\\right)\\\\[1.5ex]
&=\\frac{3}{8}a^2
\\end{align}
$$
\u3068\u306a\u308b. \u307e\u305f,
$$
\\begin{align}
\\left|\\overrightarrow{\\mathrm{BI}} \\right|^2&=\\left|\\frac{1}{4}\\left(\\overrightarrow{\\mathrm{AB}}+\\overrightarrow{\\mathrm{AC}}+\\overrightarrow{\\mathrm{AD}}\\right)-\\overrightarrow{\\mathrm{AB}} \\right|^2\\\\[1.5ex]
&=\\left|\\frac{1}{4}\\left(-3\\overrightarrow{\\mathrm{AB}}+\\overrightarrow{\\mathrm{AC}}+\\overrightarrow{\\mathrm{AD}}\\right)\\right|^2\\\\[1.5ex]
&=\\left(\\frac{1}{4}\\left(-3\\overrightarrow{\\mathrm{AB}}+\\overrightarrow{\\mathrm{AC}}+\\overrightarrow{\\mathrm{AD}}\\right)\\right)\\cdot \\left(\\frac{1}{4}\\left(-3\\overrightarrow{\\mathrm{AB}}+\\overrightarrow{\\mathrm{AC}}+\\overrightarrow{\\mathrm{AD}}\\right)\\right)\\\\[1.5ex]
&=\\frac{1}{16}\\left(9\\left|\\overrightarrow{\\mathrm{AB}} \\right|^2+\\left|\\overrightarrow{\\mathrm{AC}} \\right|^2+\\left|\\overrightarrow{\\mathrm{AD}} \\right|^2-6 \\overrightarrow{\\mathrm{AB}}\\cdot \\overrightarrow{\\mathrm{AC}}+ 2\\overrightarrow{\\mathrm{AC}}\\cdot \\overrightarrow{\\mathrm{AD}}-6\\overrightarrow{\\mathrm{AD}}\\cdot \\overrightarrow{\\mathrm{AB}} \\right)\\\\[1.5ex]
&=\\frac{1}{16}\\left(9a^2+a^2+a^2-6\\cdot\\frac{a^2}{2}+2\\cdot\\frac{a^2}{2}-6\\cdot\\frac{a^2}{2}\\right)\\\\[1.5ex]
&=\\frac{1}{16}\\cdot 6a^2\\\\[1.5ex]
&=\\frac{3}{8}a^2
\\end{align}
$$
\u3053\u306e\\(\\left|\\overrightarrow{\\mathrm{BI}}\\right|^2\\)\u306e\u8a08\u7b97\u3068\u5168\u304f\u540c\u69d8\u306b, \\(\\left|\\overrightarrow{\\mathrm{CI}}\\right|^2\\), \\(\\left|\\overrightarrow{\\mathrm{DI}}\\right|^2\\)\u3082\u8a08\u7b97\u3067\u304d, \u305d\u306e\u5024\u306f\u3044\u305a\u308c\u3082\\(\\frac{3}{8}a^2\\)\u3067\u3042\u308b.

\u3088\u3063\u3066,
$$
\\left|\\overrightarrow{\\mathrm{AI}}\\right|^2=\\left|\\overrightarrow{\\mathrm{BI}}\\right|^2=\\left|\\overrightarrow{\\mathrm{CI}}\\right|^2=\\left|\\overrightarrow{\\mathrm{DI}}\\right|^2=\\frac{3}{8}a^2
$$
\u3068\u306a\u308a,
$$
\\mathrm{AI}=\\mathrm{BI}=\\mathrm{CI}=\\mathrm{DI}\\left(=\\frac{\\sqrt{6}}{4}a\\right)
$$
\u304c\u308f\u304b\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

\u3053\u3061\u3089\u306f\u57fa\u672c\u7684\u306a\u8a08\u7b97\u3067, \u96e3\u3057\u3044\u8b70\u8ad6\u3082\u306a\u3044\u306e\u3067, \u662f\u975e\u3068\u3082\u5b8c\u7b54\u3057\u305f\u3044\u554f\u984c\u3067\u3059.

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