{"id":1714,"date":"2025-12-24T17:19:57","date_gmt":"2025-12-24T08:19:57","guid":{"rendered":"https:\/\/math-travel.com\/?p=1714"},"modified":"2026-02-11T16:54:51","modified_gmt":"2026-02-11T07:54:51","slug":"orthocenter","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-a\/orthocenter\/","title":{"rendered":"\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u3068\u306f\uff1f\u6027\u8cea\u3068\u8a3c\u660e\uff1a3\u3064\u306e\u5782\u7dda\u304c1\u70b9\u3067\u4ea4\u308f\u308b\u7406\u7531\u3092\u308f\u304b\u308a\u3084\u3059\u304f"},"content":{"rendered":"\n

\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u304c\u3044\u307e\u3044\u3061\u306a\u65b9\u306f\u5fc5\u898b\u3067\u3059\uff01<\/span><\/p>\n\n\n\n

\u4eca\u56de\u306f\u5782\u5fc3\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u500b\u5225\u6307\u5c0e\u6b7410\u5e74\u306e\u6570\u5b66\u8b1b\u5e2b\u304c\u89e3\u6c7a\u3057\u307e\u3059\u3002<\/p>\n\n\n

\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u304c\u3088\u304f\u5206\u304b\u3089\u306a\u304f\u3066\u2026<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n

\n
\"\u60a9\u3080\u5b66\u751f\"<\/figure>\n<\/div>\n\n\n

\u4e09\u89d2\u5f62\u306b\u306f\u4e94\u5fc3\u3068\u547c\u3070\u308c\u308b5\u3064\u306e\u70b9<\/span>\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u4e09\u89d2\u5f62\u306e\u4e94\u5fc3\u306e\u4e00\u89a7\"<\/figure>\n<\/div>\n\n\n

\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u3068\u306f\u3001\u5404\u9802\u70b9\u304b\u3089\u5bfe\u8fba\u306b\u5411\u304b\u3063\u3066\u5782\u7dda\u3092\u5f15\u3044\u305f\u4ea4\u70b9<\/span>\u3092\u6307\u3057\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\"<\/figure>\n<\/div>\n\n\n

\u672c\u8a18\u4e8b\u3067\u306f\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u306e\u5b9a\u7fa9\u3084\u6027\u8cea\u306b\u3064\u3044\u3066\u89e3\u8aac<\/span>\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u306b\u95a2\u3059\u308b\u60a9\u307f\u3092\u3059\u3079\u3066\u89e3\u6c7a\u3067\u304d\u308b\u3088\u3046\u306b\u89e3\u8aac\u3057\u305f\u306e\u3067\u3001\u305c\u3072\u6700\u5f8c\u307e\u3067\u898b\u3066\u3044\u3063\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u3068\u306f\uff1f<\/h2>\n\n\n\n

\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u3068\u306f\u3001\u4e09\u89d2\u5f62\u306e\u5404\u9802\u70b9\u304b\u3089\u5bfe\u8fba\u306b\u5411\u304b\u3063\u3066\u3072\u3044\u305f\u5782\u7dda\u306e\u4ea4\u70b9<\/span>\u3092\u6307\u3057\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n
\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u306e\u5b9a\u7fa9
\u5404\u9802\u70b9\u304b\u3089\u5411\u304b\u3044\u5408\u3046\u8fba\u306b\u4e0b\u3057\u305f\u5782\u7dda\u306e\u4ea4\u70b9<\/span><\/div>\n\n\n\n

\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u4e09\u89d2\u5f62\\(ABC\\)\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n

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

\u4e09\u89d2\u5f62\u306e\u5404\u9802\u70b9\u304b\u3089\u5bfe\u8fba\u306b\u5411\u304b\u3063\u3066\u5782\u7dda\u3092\u5f15\u304d\u307e\u3059\u3002<\/p>\n\n\n

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

\u3059\u308b\u30683\u672c\u306e\u5782\u7dda\u304c\uff11\u70b9\u3067\u4ea4\u308f\u308a\u3001\u305d\u306e\u70b9\u3092\u4e09\u89d2\u5f62\u306e\u5782\u5fc3<\/span>\u3068\u3088\u3073\u307e\u3059\u3002<\/p>\n\n\n

\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u5782\u7dda\u306e\u4ea4\u70b9\u3060\u304b\u3089\u5782\u5fc3\u3067\u3059\u306d\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n

\"\"\u30b7\u30fc\u30bf<\/span><\/div>
\n

\u5782\u5fc3\u306e\u5b9a\u7fa9\u306f\u5206\u304b\u308a\u3084\u3059\u304f\u3066\u3044\u3044\u306d\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u5782\u7dda\u304c1\u70b9\u3067\u4ea4\u308f\u308b\u3053\u3068\u306e\u8a3c\u660e<\/h3>\n\n\n\n

\u5782\u7dda\u304c1\u70b9\u3067\u4ea4\u308f\u308b\u3053\u3068\u306e\u8a3c\u660e\u306f\u3001\u300c2\u672c\u306e\u5782\u7dda\u306e\u4ea4\u70b9\u3092\uff13\u672c\u76ee\u306e\u5782\u7dda\u3082\u901a\u904e\u3059\u308b\u300d<\/span>\u3053\u3068\u3092\u8a3c\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u307e\u305a\u306f\uff12\u672c\u306e\u5782\u7dda\u304c\u3042\u308a\u3001\u305d\u306e\u4ea4\u70b9\u3092\u4ea4\u70b9H\u3068\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u305d\u306e\u3042\u3068\u4ea4\u70b9H\u3092\u901a\u904e\u3059\u308b\u3088\u3046\u306b\u7dda\u5206CD\u3092\u5f15\u3044\u305f\u3082\u306e\u304c\u4e0b\u56f3\u3067\u3059\u3002<\/p>\n\n\n

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

\u3053\u306e\u6bb5\u968e\u3067\u306f\u307e\u3060\\(\\angle CDA\\)\u304c\u5782\u76f4\u3068\u306f\u5206\u304b\u308a\u307e\u305b\u3093\u3002<\/p>\n\n\n\n

\u305d\u3053\u3067\u3001\u7dda\u5206CD\u304c\u5782\u76f4\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3067\u304d\u308c\u3070\u30013\u672c\u306e\u5782\u7dda\u304c\uff11\u70b9\u3067\u4ea4\u308f\u308b\u3053\u3068\u3092\u793a\u305b\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\\(\\angle CFH,\\angle CEH\\)\u306f\u3068\u3082\u306b\u76f4\u89d2\u306a\u306e\u3067\u3001 C, F, H, E \u306e4\u70b9\u306f\u3001CH\u3092\u76f4\u5f84\u3068\u3059\u308b\u5186\u5468\u4e0a\u306b\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u306e\u3053\u3068\u304b\u3089\u3001\u5186\u5468\u89d2\u306e\u5b9a\u7406\u3088\u308a<\/p>\n\n\n\n

\\(\\angle HCF=\\angle HEF\\)\u304c\u8a00\u3048\u307e\u3059\u3002(\u4e0b\u56f3\u3092\u53c2\u8003\u306b\u3057\u3066\u304f\u3060\u3055\u3044)<\/p>\n\n\n

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

\u307e\u305f\u3001\\(\\angle AEB,\\angle AFB\\)\u3082\u3068\u3082\u306b\u76f4\u89d2\u306a\u306e\u3067\u3001 A, B, E, F \u306e4\u70b9\u306f\u3001AB\u3092\u76f4\u5f84\u3068\u3059\u308b\u5186\u5468\u4e0a\u306b\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u306e\u3053\u3068\u304b\u3089\u3001\\(\\angle AEF=\\angle ABF\\)\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n

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

\u3088\u3063\u3066\u3001<\/p>\n\n\n\n

2\u3064\u3092\u5408\u308f\u305b\u308b\u3068\u3001\\(\\angle ACD=\u2220ABF\\)\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\\begin{eqnarray}
\\angle \\mathrm{ADC} &=&180^{\\circ}-\\angle \\mathrm{DAC}-\\angle \\mathrm{ACD} \\\\
&=&180^{\\circ}-\\angle \\mathrm{BAF}-\\angle \\mathrm{ABF} \\\\
&=&\\angle \\mathrm{AFB} \\\\
&=&90^{\\circ}
\\end{eqnarray}<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u3001\uff13\u672c\u306e\u5782\u7dda\u304c\uff11\u70b9\u3067\u4ea4\u308f\u308b\u3053\u3068\u304c\u8a3c\u660e\u3055\u308c\u307e\u3057\u305f\u3002<\/p>\n\n\n

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

\u5782\u5fc3\u304c\u6301\u3064\u6027\u8cea<\/h2>\n\n\n\n

\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u306b\u306f\uff12\u3064\u306e\u6027\u8cea\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u307e\u305a\u6027\u8cea\u3092\u7d39\u4ecb\u3057\u3066\u304b\u3089\u8a3c\u660e\u3092\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\uff11\uff1a\u56db\u89d2\u5f62ADHF,BEHD,CFHE\u306f\u5186\u306b\u5185\u63a5\u3059\u308b\u56db\u89d2\u5f62\u3067\u3042\u308b<\/span><\/p>\n\n\n

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

\uff12\uff1a\\(AH=2R\\cos A\\)<\/span>
R\u306f\u4e09\u89d2\u5f62ABC\u306e\u5916\u63a5\u5186\u306e\u534a\u5f84\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n

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

\u5782\u5fc3\u306e\u6027\u8cea\u300a\u8a3c\u660e\u300b<\/h2>\n\n\n\n

\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u304c\u3082\u3064\u6027\u8cea\u306e\u8a3c\u660e\u3092\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u307e\u305a\u306f\u300c\u56db\u89d2\u5f62ADHF,BEHD,CFHE\u306f\u5186\u306b\u5185\u63a5\u3059\u308b\u56db\u89d2\u5f62\u3067\u3042\u308b<\/span>\u300d\u3092\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u308c\u306b\u306f\u3001\u4e2d\u5b663\u5e74\u751f\u3067\u7fd2\u3063\u305f\u5186\u5468\u89d2\u306e\u5b9a\u7406<\/span>\u3092\u4f7f\u3046\u3068\u7406\u89e3\u304c\u65e9\u3044\u3067\u3059\u3002<\/p>\n\n\n\n

\u56db\u89d2\u5f62CFHE\u306b\u6ce8\u76ee\u3059\u308b\u3068\u3001\\(\\angle CFH=90^\\circ\\),\\(\\angle CEH=90^\\circ\\)<\/p>\n\n\n

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

\u3057\u305f\u304c\u3063\u3066\u3001\u5186\u5468\u89d2\u306e\u5b9a\u7406\u306e\u9006\u3092\u7528\u3044\u3066\u3001CH\u3092\u76f4\u5f84\u3068\u3059\u308b\u5186\u304c\u898b\u3048\u3066\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u4ed6\u306e\u56db\u89d2\u5f62\u3067\u3082\u540c\u69d8\u306e\u8a3c\u660e\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u3064\u304e\u306b\u5782\u5fc3\u306e\u6027\u8cea\uff12\u300c\\(AH=2R\\cos A\\)<\/span>\u300d\u306e\u8a3c\u660e\u3092\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u308c\u306f\u4e09\u89d2\u95a2\u6570\u306e\u6b63\u5f26\u5b9a\u7406\u3092\u7528\u3044\u3066\u8a3c\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u4e09\u89d2\u5f62ABC\u306e\u5916\u63a5\u5186\u306e\u534a\u5f84\u3092\\(R\\)\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n

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

\\(AH=\\displaystyle \\frac{AF}{\\cos \\angle CAE}\\)<\/p>\n\n\n\n

\\(AF=AB\\cos A\\)<\/p>\n\n\n\n

\\(\\cos \\angle CAE=\\sin(90^\\circ-\\angle CAE)=\\sin \\angle ACE\\)<\/p>\n\n\n\n

\u306a\u306e\u3067\u3001<\/p>\n\n\n\n

\\[\\displaystyle \\frac{AF}{\\cos \\angle CAE}=\\displaystyle \\frac{AB\\cos A}{\\sin \\angle ACE}\\]<\/p>\n\n\n

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

\u4e09\u89d2\u5f62ABC\u306e\u5916\u63a5\u5186\u306e\u534a\u5f84\u3092R\u3068\u3057\u305f\u3068\u304d\u3001<\/p>\n

\\[\\displaystyle \\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}=2R\\]<\/p>\n<\/div>\n\n\n\n

\u6b63\u5f26\u5b9a\u7406\u306b\u3064\u3044\u3066\u77e5\u308a\u305f\u3044\u65b9\u306f\u300c\u6b63\u5f26\u5b9a\u7406\u306e\u516c\u5f0f\u3068\u8a3c\u660e\u300d\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u6b63\u5f26\u5b9a\u7406\u3088\u308a<\/p>\n\n\n\n

\\[\\displaystyle \\frac{AB\\cos A}{\\sin \\angle ACE}=2R\\cos A\\]<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n

\\[AH=2R\\cos A\\]<\/p>\n\n\n\n

\u8a3c\u660e\u7d42\u4e86\u3002<\/p>\n\n\n\n

\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u306e\u898b\u3064\u3051\u65b9<\/h2>\n\n\n
\n
\"\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u306e\u898b\u3064\u3051\u65b9\"<\/figure>\n<\/div>\n\n\n

\u4e09\u89d2\u5f62\u306e\u5782\u5fc3\u306f\u4f5c\u56f3\u3060\u3051\u3067\u306a\u304f\u3001\u5ea7\u6a19\u3084\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u3092\u6c42\u3081\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\n