{"id":13722,"date":"2025-12-24T17:19:57","date_gmt":"2025-12-24T08:19:57","guid":{"rendered":"https:\/\/math-travel.com\/?p=13722"},"modified":"2026-02-11T16:43:27","modified_gmt":"2026-02-11T07:43:27","slug":"cevas-theorem","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-a\/cevas-theorem\/","title":{"rendered":"\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306e\u4f7f\u3044\u65b9\u3068\u8a3c\u660e\uff01\u8fba\u306e\u6bd4\u3092\u6c42\u3081\u308b\u30b3\u30c4\u3092\u30e1\u30cd\u30e9\u30a6\u30b9\u3068\u306e\u9055\u3044\u3068\u5171\u306b\u89e3\u8aac"},"content":{"rendered":"\n
\u300c\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u3063\u3066\u3069\u3093\u306a\u516c\u5f0f\uff1f\u300d
\u300c\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306e\u4f7f\u3044\u65b9\u304c\u77e5\u308a\u305f\u3044\u300d<\/p>\n<\/div><\/div>\n\n\n\n
\u4eca\u56de\u306f\u6570\u5b661A\u306e\u56f3\u5f62\u306e\u6027\u8cea\u304b\u3089\u300c\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u300d<\/span>\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002<\/p>\n\n\n \u56f3\u5f62\u306e\u554f\u984c\u306f\u3069\u3046\u3057\u3066\u3082\u82e6\u624b\u3067\u2026<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n \u25b3ABC\u306e\u5185\u90e8\u306b\u70b9O\u304c\u3042\u308b\u3002 \\[\\displaystyle \\frac{BP}{PC} \\cdot \\frac{CQ}{QA} \\cdot \\frac{AR}{RB}=1\\]<\/p>\n<\/div><\/div>\n\n\n\n \u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306f\u5404\u8fba\u306e\u6bd4\u3092\u7528\u3044\u3066\u3001\u4e09\u89d2\u5f62\u306e\u8fba\u306e\u9577\u3055\u3092\u6c42\u3081\u308b\u516c\u5f0f\u3067\u3059\u3002<\/p>\n\n\n\n \u672c\u8a18\u4e8b\u3067\u306f\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306e\u4f7f\u3044\u65b9\u3084\u8a3c\u660e\u65b9\u6cd5\u3001\u899a\u3048\u65b9\u306a\u3069\u306b\u3064\u3044\u3066\u89e3\u8aac<\/span>\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \u307e\u305f\u3001\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u3068\u30bb\u30c3\u30c8\u3067\u7fd2\u3046\u300c\u30e1\u30cd\u30e9\u30a6\u30b9\u306e\u5b9a\u7406\u300d<\/span>\u306b\u3064\u3044\u3066\u3082\u8aac\u660e\u3057\u3066\u3044\u308b\u306e\u3067\u3001\u5408\u308f\u305b\u3066\u53c2\u8003\u306b\u3057\u3066\u3044\u305f\u3060\u3051\u308c\u3070\u3068\u601d\u3044\u307e\u3059\uff01<\/p>\n\n\n \u5165\u587e\u3092\u691c\u8a0e\u3057\u3066\u3044\u308b\u65b9\u306f \u305d\u308c\u3067\u306f\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n \u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306f\u8fba\u306e\u6bd4\u3084\u9577\u3055\u3092\u6c42\u3081\u308b\u6642\u306b\u4f7f\u3046\u5b9a\u7406\u3067\u3059\u3002<\/p>\n\n\n\n \u25b3ABC\u306e\u5185\u90e8\u306b\u70b9O\u304c\u3042\u308b\u3002 \\[\\displaystyle \\frac{BP}{PC} \\cdot \\frac{CQ}{QA} \\cdot \\frac{AR}{RB}=1\\]<\/p>\n<\/div><\/div>\n\n\n\n \u4e0a\u8a18\u306e\u516c\u5f0f\u306b\u3057\u305f\u304c\u3063\u3066\u3001\u5404\u8fba\u306e\u5024\u3092\u5f0f\u306b\u4ee3\u5165\u3057\u3066\u7a4d\u3092\u6c42\u3081\u308b\u30681\u306b\u306a\u308b\u3068\u3044\u3046\u5b9a\u7406\u3067\u3059\u306d\u3002<\/p>\n\n\n\n \u3053\u3053\u304b\u3089\u306f\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306e\u8a3c\u660e\u65b9\u6cd5\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \u307e\u305a\u3001\u25b3ABC\u306b\u304a\u3044\u3066\u25b3OAB\u3068\u25b3OAC\u306b\u6ce8\u76ee\u3057\u3066\u8003\u3048\u307e\u3059\u3002<\/p>\n\n\n \u3059\u308b\u3068\u3001\u3053\u306e\u3068\u304d\u6b21\u306e\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n \\[\\displaystyle \\frac{\\triangle{OAB}}{\\triangle{OCA}}=\\frac{BP}{PC}\\]<\/p>\n\n\n\n \u307e\u305a\u306f\u3001\u3053\u308c\u3092\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \u3010\u8a3c\u660e\u3011 \u6b21\u306b\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u3092\u8003\u3048\u308b\u3068\u3001<\/p>\n\n\n\n \\(\\displaystyle \\triangle{OAB}=AO \\times BH \\times \\frac{1}{2}\\)\u2015\u2461 \u2461,\u2462\u3067AO\u3068\\(\\frac{1}{2}\\)\u3092\u6d88\u53bb\u3059\u308b\u3068\u3001<\/p>\n\n\n\n \\[\\triangle{OAB}\uff1a\\triangle{OAC}\uff1dBH\uff1aCK\u2015\u2463\\]<\/p>\n\n\n\n \u2463\u3088\u308a\u3001<\/p>\n\n\n\n \\[BH:CK=BP:PC\\]<\/p>\n\n\n\n \u2463,\u2464\u3088\u308a\u3001\\(\\triangle\\ OAB\uff1a\\triangle OAC\uff1dBP:PC\\)<\/p>\n\n\n\n \u3053\u308c\u3067\u3001<\/p>\n\n\n\n \\[\\displaystyle \\frac{\\triangle{OAB}}{\\triangle{OCA}}=\\frac{BP}{PC}\\]<\/p>\n\n\n\n \u304c\u8a3c\u660e\u3055\u308c\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \u3053\u308c\u3068\u540c\u3058\u3088\u3046\u306b\u8003\u3048\u308b\u3068\u3001<\/p>\n\n\n\n \\(\\displaystyle \\frac{\\triangle{OAB}}{\\triangle{OCA}}=\\frac{CQ}{QA}\\)\u2015\u2466<\/p>\n\n\n\n \\(\\displaystyle \\frac{\\triangle{OBC}}{\\triangle{OAB}}=\\frac{AR}{RB}\\)\u2015\u2467<\/p>\n\n\n\n \u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \u2465,\u2466,\u2467\u3088\u308a\u3001<\/p>\n\n\n\n \\begin{eqnarray} \u3068\u306a\u308a\u3001\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u304c\u8a3c\u660e\u3055\u308c\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \u3053\u3053\u307e\u3067\u3001\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306e\u8a3c\u660e\u65b9\u6cd5\u3092\u8aac\u660e\u3057\u3066\u304d\u307e\u3057\u305f\u3002 \u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u3092\u899a\u3048\u308b\u3046\u3048\u3067\u306e\u30dd\u30a4\u30f3\u30c8\u306f\u3001\u56f3\u306e\u3088\u3046\u306b\u300c\u4e09\u89d2\u5f62\u306e1\u3064\u306e\u9802\u70b9\u304b\u3089\u30011\u5468\u3059\u308b\u3088\u3046\u306b\u30a2\u30eb\u30d5\u30a1\u30d9\u30c3\u30c8\u3092\u305f\u3069\u3063\u3066\u3044\u304f\u300d<\/span>\u3053\u3068\u3067\u3059\u3002 \u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306f\u30a2\u30eb\u30d5\u30a1\u30d9\u30c3\u30c8\u306e\u9806\u756a\u304c\u3068\u3066\u3082\u5927\u4e8b\u3067\u3059\u3002 \u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u3067\u3059\u304c\u3001\u5b9f\u969b\u3069\u3093\u306a\u3068\u304d\u306b\u4f7f\u3046\u306e\u304b\u5206\u304b\u3089\u306a\u3044\u4eba\u3082\u3044\u307e\u3059\u3088\u306d\u3002<\/p>\n\n\n\n \u3053\u3053\u304b\u3089\u306f\u3001\u3044\u3064\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u3092\u4f7f\u3048\u3070\u826f\u3044\u306e\u304b\u3001\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002 \u3053\u306e\u3068\u304d\u3001\u5bfe\u8fba\u306b\u5411\u304b\u3063\u3066\u5f15\u3044\u305f\u7dda\u306f\u5782\u7dda\u3084\u89d2\u306e\u4e8c\u7b49\u5206\u7dda\u3067\u3042\u308b\u5fc5\u8981\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n\n\n\n \u3068\u306b\u304b\u304f3\u3064\u306e\u7dda\u304c1\u70b9\u3067\u4ea4\u308f\u3063\u3066\u3044\u308b\u3068\u304d\u306b\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306f\u767a\u52d5\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \u3053\u3053\u304b\u3089\u306f\u3001\u5b9f\u969b\u306b\u7df4\u7fd2\u554f\u984c\u3092\u3084\u3063\u3066\u307f\u307e\u3057\u3087\u3046\uff01<\/p>\n\n\n\n (1) \u6b21\u306e\u56f3\u306b\u304a\u3044\u3066AR\u306e\u5024\u3092\u6c42\u3081\u306a\u3055\u3044\u3002<\/p>\n\n\n (2) \u6b21\u306e\u56f3\u306b\u304a\u3044\u3066AR\uff1aRB\u3092\u6c42\u3081\u306a\u3055\u3044\u3002<\/p>\n\n\n \u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u3088\u308a\u3001<\/p>\n\n\n\n \\begin{eqnarray} \u3057\u305f\u304c\u3063\u3066\u3001\u3000AR\uff1d4<\/span><\/p>\n\n\n\n \u3053\u306e\u554f\u984c\u3067\u306f\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u3092\u6642\u8a08\u56de\u308a\u3067\u8003\u3048\u307e\u3057\u305f\u304c\u3001\u53cd\u6642\u8a08\u56de\u308a\u3067\u8003\u3048\u3066\u3082\u826f\u3044\u3067\u3059\u3002<\/p>\n\n\n\n \u3067\u3059\u304c\u3001\u5f0f\u3092\u7acb\u3066\u305f\u6642\u306b\u3001\u6c42\u3081\u308b\u30a2\u30eb\u30d5\u30a1\u30d9\u30c3\u30c8\u306f\u5206\u5b50\u306b\u3042\u3063\u305f\u65b9\u304c\u8a08\u7b97\u3057\u3084\u3059\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n \u4eca\u56de\u306e\u554f\u984c\u306e\u5834\u5408\u306f\u53cd\u6642\u8a08\u56de\u308a\u3067\u8003\u3048\u3066\u3057\u307e\u3046\u3068\u3001\u6c42\u3081\u308bAR\u304c\u5206\u6bcd\u306b\u306a\u3063\u3066\u3057\u307e\u3044\u3001\u8a08\u7b97\u3057\u306b\u304f\u304f\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \\[\\frac{2}{AR} \\cdot \\frac{3}{1} \\cdot \\frac{2}{3}\uff1d1\\]<\/p>\n\n\n\n
\u9ad8\u6821\u751f<\/span><\/div>
<\/figure>\n<\/div>\n\n\n
\u9802\u70b9A,B,C\u3068\u70b9O\u3092\u7d50\u3076\u76f4\u7dda\u304c\u3001\u5411\u304b\u3044\u5408\u3046\u8fba\u3068\u305d\u308c\u305e\u308c\u70b9P,Q,R\u3067\u4ea4\u308f\u308b\u3068\u304d<\/p>\n\n\n\n
\u30b7\u30fc\u30bf<\/span><\/div>
\u305c\u3072\u6700\u5f8c\u307e\u3067\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u3068\u306f<\/h2>\n\n\n\n
<\/figure>\n<\/div>\n\n\n
\u9802\u70b9A,B,C\u3068\u70b9O\u3092\u7d50\u3076\u76f4\u7dda\u304c\u3001\u5411\u304b\u3044\u5408\u3046\u8fba\u3068\u305d\u308c\u305e\u308c\u70b9P,Q,R\u3067\u4ea4\u308f\u308b\u3068\u304d<\/p>\n\n\n\n\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306e\u8a3c\u660e\u65b9\u6cd5<\/h2>\n\n\n\n
<\/figure>\n<\/div>\n\n\n
\u70b9B,C\u304b\u3089\u5782\u76f4\u306b\u4e0b\u3057\u305f\u70b9\u3092\u305d\u308c\u305e\u308cH,K\u3068\u3059\u308b\u3068\u3001\\(BH \/\/ CK\\)\u2015\u2460<\/p>\n\n\n\n
\\(\\displaystyle \\triangle{OAC}=AO \\times CK \\times \\frac{1}{2}\\)\u2015\u2462<\/p>\n\n\n\n
&&\\frac{BP}{PC} \\cdot \\frac{CQ}{QA} \\cdot \\frac{AR}{RB}\\\\
&=&\\frac{\\triangle{OAB}}{\\triangle{OCA}} \\cdot \\frac{\\triangle{OBC}}{\\triangle{OAB}} \\cdot \\frac{\\triangle{OCA}}{\\triangle{ABC}}\\\\
&=&1
\\end{eqnarray}<\/p>\n\n\n\n\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306e\u899a\u3048\u65b9<\/h2>\n\n\n\n
\u3053\u3053\u304b\u3089\u306f\u3001\u300c\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u3001\u899a\u3048\u3089\u308c\u306a\u3044\uff01\u300d\u3068\u3044\u3046\u4eba\u306b\u5411\u3051\u3066\u899a\u3048\u65b9\u3092\u3054\u7d39\u4ecb\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n
\u4e0b\u306e\u56f3\u3067\u8aac\u660e\u3059\u308b\u3068\u3001\u70b9\uff21\u304b\u3089\u30b9\u30bf\u30fc\u30c8\u3057\u3066\u70b9\uff21\u306b\u623b\u308b\u3088\u3046\u306b\u3057\u30661\u5468\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
<\/figure>\n<\/div><\/div><\/div>\n\n\n\n
<\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n
\u305d\u306e\u305f\u3081\u3001\u9806\u756a\u3092\u9593\u9055\u3048\u308b\u3068\u7b54\u3048\u304c\u5408\u308f\u306a\u304f\u306a\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u3002
\u4e0b\u56f3\u306e\u2192\u306e\u9806\u756a\u3067\u5f0f\u3092\u7acb\u3066\u3066\u3044\u304f\u3068\u3001\u9593\u9055\u3048\u306b\u304f\u3044\u306e\u3067\u306f\u306a\u3044\u304b\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306e\u4f7f\u3044\u65b9<\/h2>\n\n\n\n
\u307e\u305a\u3001\u65e9\u901f\u3067\u3059\u304c\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306f\u300c\u4e09\u89d2\u5f62\u306e3\u3064\u306e\u9802\u70b9\u304b\u3089\u5bfe\u8fba\u306b\u5411\u304b\u3063\u3066\u5f15\u3044\u305f3\u672c\u306e\u7dda\u304c\uff11\u70b9\u3067\u4ea4\u308f\u308b\u3068\u304d\u300d<\/span>\u306b\u4f7f\u3044\u307e\u3059\u3002<\/p>\n\n\n
<\/figure>\n<\/div>\n\n\n\u30c1\u30a7\u30d0\u306e\u5b9a\u7406\u306e\u7df4\u7fd2\u554f\u984c<\/h2>\n\n\n\n
<\/figure>\n<\/div>\n\n\n
<\/figure>\n<\/div>\n\n\n\u7df4\u7fd2\u554f\u984c\u2460\u306e\u89e3\u7b54<\/h3>\n\n\n\n
\\displaystyle \\frac{QC}{AQ} \\cdot \\frac{PB}{CP} \\cdot \\frac{RA}{BR}&=&1\\\\
\\displaystyle \\frac{3}{2} \\cdot \\frac{1}{3} \\cdot \\frac{AR}{2}&\uff1d&1\\\\
AR&\uff1d&4
\\end{eqnarray}<\/p>\n\n\n\n\u2606\u7df4\u7fd2\u554f\u984c\u2460\u306e\u89e3\u8aac<\/h4>\n\n\n
<\/figure>\n<\/div>\n\n\n