{"id":1674,"date":"2025-12-24T17:19:57","date_gmt":"2025-12-24T08:19:57","guid":{"rendered":"https:\/\/math-travel.com\/?p=1674"},"modified":"2026-02-11T16:42:36","modified_gmt":"2026-02-11T07:42:36","slug":"circumcenter","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-a\/circumcenter\/","title":{"rendered":"\u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u3068\u306f\uff1f\u6027\u8cea\u3068\u8a3c\u660e\u30fc\u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u306e\u4ea4\u70b9\u306b\u306a\u308b\u7406\u7531\u3092\u30b9\u30c3\u30ad\u30ea\u89e3\u8aac"},"content":{"rendered":"\n
\u300c\u5916\u5fc3\u306e\u6027\u8cea\u3063\u3066\u306a\u3093\u3060\u3063\u3051\uff1f\u300d<\/span> \u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u3063\u3066\u3069\u3093\u306a\u70b9\u3060\u3063\u3051\u2026<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/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 \u4eca\u56de\u89e3\u8aac\u3059\u308b\u300c\u5916\u5fc3\u300d\u306f\u4e94\u5fc3\u306e\u91cd\u8981\u306a\u70b9\u306e1\u3064<\/span>\u3067\u3059\u3002<\/p>\n\n\n\n \u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u3068\u306f\u3001\u300c\u4e09\u89d2\u5f62\u306e\u5404\u8fba\u306e\u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u306e\u4ea4\u70b9<\/span>\u300d\u3092\u6307\u3057\u307e\u3059\u3002<\/p>\n\n\n \u4e09\u89d2\u5f62\u306e\u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u3092\u5f15\u304f\u3068\u30011\u70b9\u3067\u4ea4\u308f\u308b\u306e\u3067\u3059\u3002<\/p>\n\n\n\n \u307e\u305f\u3001\u5916\u5fc3\u306f\u4e09\u89d2\u5f62\u306e\u5916\u63a5\u5186\u306e\u4e2d\u5fc3<\/span>\u3067\u3059\u3002<\/p>\n\n\n \u672c\u8a18\u4e8b\u3067\u306f\u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u306e\u5b9a\u7fa9\u3084\u6027\u8cea\u3092\u89e3\u8aac<\/span>\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \u5916\u5fc3\u306e\u7591\u554f\u304c\u3042\u308b\u65b9\u306f\u305c\u3072\u6700\u5f8c\u307e\u3067\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n \u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u3068\u306f\u3001\u4e09\u89d2\u5f62\u306e\u5404\u8fba\u306e\u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u306e\u4ea4\u70b9<\/span>\u3092\u6307\u3057\u307e\u3059\u3002<\/p>\n\n\n \u4e0b\u56f3\u306e\u3088\u3046\u306a\u4e09\u89d2\u5f62\\(ABC\\)\u304c\u3042\u308b\u3068\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n \u3053\u306e\u4e09\u89d2\u5f62\u306e\u3059\u3079\u3066\u306e\u8fba\u306b\u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u3092\u5f15\u304d\u307e\u3059\u3002<\/p>\n\n\n \u3059\u308b\u3068\u30013\u672c\u306e\u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u304c\uff11\u70b9\u3067\u4ea4\u308f\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \u3053\u306e\u70b9\u3092\u4e09\u89d2\u5f62\u306e“\u5916\u5fc3”<\/span>\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n \u307e\u305f\u3001\u4e09\u89d2\u5f62\u306e\u5404\u9802\u70b9\u3092\u901a\u308b\u5186\u3092\u300c\u5916\u63a5\u5186\u300d<\/span>\u3068\u3044\u3044\u307e\u3059\u3002<\/p>\n\n\n \u5916\u5fc3\u306b\u306f\u300c\u5916\u63a5\u5186\u306e\u4e2d\u5fc3\u300d<\/span>\u3068\u3044\u3046\u6027\u8cea\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n \u5404\u8fba\u306e\u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u3067\u5916\u5fc3\u3092\u898b\u3064\u3051\u3089\u308c\u308b\u3088<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n \u5916\u5fc3\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n \\(\\triangle ABC\\)\u306e\u9802\u70b9\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u3092\\(\\mathrm{A}(\\vec{a}), \\mathrm{B}(\\vec{b}), \\mathrm{C}(\\vec{c})\\)\u3068\u3057\u3066\u3001\u5fc3\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u3092\\(P(\\vec{p})\\)\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n \\[\\displaystyle \\vec{p}=\\frac{\\vec{a} \\sin 2 A+\\vec{b} \\sin 2 B+\\vec{c} \\sin 2 C}{\\sin 2 A+\\sin 2 B+\\sin 2 C}\\]<\/p>\n<\/div><\/div>\n\n\n\n \u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u3092\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n \u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u306e\u516c\u5f0f\u3092\u8a3c\u660e\u3059\u308b\u306b\u3042\u305f\u3063\u3066\u3001\u78ba\u8a8d\u3057\u3066\u304a\u304d\u305f\u3044\u3053\u3068\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \u4e09\u89d2\u5f62\\(\\triangle \\mathrm{ABC}(\\mathrm{A}(\\vec{a}), \\mathrm{B}(\\vec{b}), \\mathrm{C}(\\vec{c}))\\)\u306e\u5185\u90e8\u306b\u70b9 \\(\\mathrm{J}(\\vec{j})\\)\u3092\u3068\u308b\u3002<\/p>\n \u3053\u306e\u6642\\(\\triangle \\mathrm{JBC},\\triangle \\mathrm{JCA},\\triangle \\mathrm{JAB}\\)\u3068\u3057\u3066\u305d\u308c\u305e\u308c\u306e\u9762\u7a4d\u6bd4\u3092<\/p>\n \\(\\triangle \\mathrm{JBC}: \\triangle \\mathrm{JCA}: \\triangle \\mathrm{JAB}=p: q: r \\quad(p, q, r>0)\\)\u3068\u3059\u308b\u3002<\/p>\n \u3053\u306e\u3068\u304d<\/p>\n \u304c\u6210\u308a\u7acb\u3064\u3002<\/p>\n<\/div>\n\n\n\n \u3053\u308c\u3092\u6d3b\u7528\u3057\u3066\u5916\u5fc3\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u306e\u516c\u5f0f\u3092\u8a3c\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \\(\\triangle ABC\\)\u306e\u5916\u63a5\u5186\u306e\u4e2d\u5fc3\u3092\u70b9\\(P\\)\u3001\u534a\u5f84\u3092\\(R\\)\u3068\u3059\u308b\u3002<\/p>\n\n\n \u3053\u306e\u3068\u304d\u3001\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u6bd4\u306f<\/p>\n\n\n\n \\begin{eqnarray} \u3053\u3053\u3067\u5148\u307b\u3069\u78ba\u8a8d\u3057\u305f\u4e09\u89d2\u5f62\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u3068\u9762\u7a4d\u6bd4\u306e\u516c\u5f0f\u3088\u308a\u3001<\/p>\n\n\n\n \\(p=\\sin 2 A, q=\\sin 2 B,r=\\sin 2 C\\)\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n \\[\\displaystyle \\vec{p}=\\frac{\\vec{a} \\sin 2 A+\\vec{b} \\sin 2 B+\\vec{c} \\sin 2 C}{\\sin 2 A+\\sin 2 B+\\sin 2 C}\\]<\/p>\n\n\n\n \u3053\u308c\u3067\u3001\u8a3c\u660e\u7d42\u4e86\u3067\u3059\u3002<\/p>\n\n\n \u96e3\u3057\u304f\u3066\u3088\u304f\u5206\u304b\u3089\u306a\u304b\u3063\u305f\u3067\u3059\u2026<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n \u96e3\u3057\u3044\u3068\u3053\u308d\u306a\u306e\u3067\u3001\u3072\u3068\u307e\u305a\u98db\u3070\u3057\u3066\u3082\u5927\u4e08\u592b\u3067\u3059<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n \u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u306b\u306f\u3001\u3044\u304f\u3064\u304b\u306e\u6027\u8cea\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \uff11\uff1a\u5404\u8fba\u306b\u5782\u7dda\u3092\u5f15\u304f\u30682\u7b49\u5206\u3059\u308b<\/span><\/p>\n\n\n \uff12\uff1a\u5404\u9802\u70b9\u304b\u3089\u306e\u8ddd\u96e2\u304c\u7b49\u3057\u3044<\/span><\/p>\n\n\n \uff13\uff1a\u5916\u63a5\u5186\u306e\u4e2d\u5fc3<\/span><\/p>\n\n\n \u3053\u308c\u306f\uff12\u306e\u6027\u8cea\u3088\u308a\u3001\u5404\u9802\u70b9\u307e\u3067\u306e\u8ddd\u96e2\u3092\u534a\u5f84\u3068\u3059\u308b\u3053\u3068\u3067\u5186\u304c\u898b\u3048\u3066\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u3001\u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u306f\u5916\u63a5\u5186\u306e\u4e2d\u5fc3\u3067\u3082\u3042\u308b\u306e\u3067\u3059\u3002<\/span><\/p>\n\n\n\n ✅\u3000\u4e09\u89d2\u5f62\u306e\u4e94\u5fc3\u307e\u3068\u3081\u8a18\u4e8b<\/span><\/p>\n\n\n\n \u4e09\u89d2\u5f62\u306e\u4e94\u5fc3\uff08\u5185\u5fc3\u3001\u5916\u5fc3\u3001\u91cd\u5fc3\u3001\u5782\u5fc3\u3001\u508d\u5fc3\uff09\u6027\u8cea\u307e\u3068\u3081<\/p>\n\n\n\n \u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u304c\u6301\u3064\u6027\u8cea\u3092\u306a\u305c\u305d\u3046\u306a\u308b\u306e\u304b\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \uff11\u306e\u300c\u5404\u8fba\u306b\u5782\u7dda\u3092\u5f15\u304f\u30682\u7b49\u5206\u3059\u308b\u300d\u306f\u3001\u5916\u5fc3\u306e\u5b9a\u7fa9\u300c\u5404\u8fba\u306e\u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u306e\u4ea4\u70b9\u300d\u306e\u8a00\u3044\u63db\u3048<\/span>\u3067\u3059\u306d\u3002<\/p>\n\n\n\n \u300c\u5916\u5fc3\u304b\u3089\u5782\u7dda\u3092\u5f15\u304f\u30682\u7b49\u5206\u3059\u308b\u300d \u3053\u306e\u3069\u3061\u3089\u3068\u3082\u3044\u3048\u307e\u3059\u3002<\/p>\n\n\n\n \u3064\u304e\u306b\uff12\u300c\u5404\u9802\u70b9\u304b\u3089\u306e\u8ddd\u96e2\u304c\u7b49\u3057\u3044\u300d\u3092\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \u5916\u5fc3\u3068\uff12\u3064\u306e\u9802\u70b9\u3092\u7d50\u3076\u3068\u4e09\u89d2\u5f62\u304c\u898b\u3048\u3066\u304d\u307e\u3059\u3002<\/p>\n\n\n \u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u3092\u5f15\u304f\u3053\u3068\u3067\u3001\u4e09\u89d2\u5f62\u304c\uff12\u3064\u306b\u5206\u3051\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n \u3053\u306e\uff12\u3064\u306e\u4e09\u89d2\u5f62\u306f\u3001\uff12\u8fba\u3068\u305d\u306e\u9593\u306e\u89d2\u304c\u305d\u308c\u305e\u308c\u7b49\u3057\u3044\u306e\u3067\u3001\u5408\u540c\u306b\u306a\u308a\u307e\u3059\u3002<\/span><\/p>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u3001\\(OA=OB\\)<\/p>\n\n\n \u4ed6\u306e\u9802\u70b9\u3067\u3082\u3001\u540c\u69d8\u306b\u4e8c\u7b49\u8fba\u4e09\u89d2\u5f62\u306b\u306a\u308a\u3001\\(OA=OC\\)<\/p>\n\n\n \u3088\u3063\u3066\u3001\\(OA=OB=OC\\)\u3068\u306a\u308a\u3001\u5916\u5fc3\u306f\u5404\u9802\u70b9\u304b\u3089\u306e\u8ddd\u96e2\u304c\u7b49\u3057\u3044\u3053\u3068\u304c\u8a3c\u660e\u3055\u308c\u307e\u3057\u305f\u3002<\/p>\n\n\n \uff13\u300c\u5916\u63a5\u5186\u306e\u4e2d\u5fc3\u300d\u306f\uff12\u306e\u6027\u8cea\u3092\u5fdc\u7528\u3059\u308b\u3068\u3001\u7c21\u5358\u306b\u8a3c\u660e\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \uff12\u306e\\(OA=OB=OC\\)\u3088\u308a\uff13\u70b9\u3059\u3079\u3066\u7b49\u8ddd\u96e2\u306b\u3042\u308b\u306e\u3067\u3001\u5916\u5fc3\u3092\u5186\u306e\u4e2d\u5fc3\u3068\u3057\u3066\u3001\u5186\u3092\u63cf\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n \u3057\u305f\u304c\u3063\u3066\u3001\u5916\u5fc3\u306f\u5916\u63a5\u5186\u306e\u4e2d\u5fc3\u3067\u3042\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n “\u5185\u5fc3”<\/span>\u3068“\u5916\u5fc3”<\/span>\u306f\u3088\u304f\u51fa\u3066\u304f\u308b\u70b9\u306a\u306e\u3067\u3001\u9055\u3044\u3092\u7406\u89e3\u3057\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n \u5916\u5fc3\u306f\u4e09\u89d2\u5f62\u306e\u5916\u63a5\u5186\u306e\u4e2d\u5fc3\u3067\u3059\u304c\u3001\u5185\u5fc3\u306f\u5185\u63a5\u5186\u306e\u4e2d\u5fc3\u3068\u306a\u308b\u70b9<\/span>\u3067\u3059\u3002<\/p>\n\n\n \u307e\u305f\u3001\u5185\u5fc3\u306f\u89d2\u306e\u4e8c\u7b49\u5206\u7dda\u306e\u4ea4\u70b9\u3067\u3059\u3002<\/p>\n\n\n \u4e09\u89d2\u5f62\u306e\u5185\u5fc3\u306e\u6027\u8cea\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \u5185\u5fc3\u306b\u3064\u3044\u3066\u306f\u300c \u4e09\u89d2\u5f62\u306e\u5185\u5fc3\u306e\u6027\u8cea\u3068\u8a3c\u660e\u300d\u3067\u8a73\u3057\u304f\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059\u3002<\/span><\/p>\n\n\n \u5916\u5fc3\u306f\u4e2d\u7dda\u306e\u4ea4\u70b9\u306a\u3093\u3067\u3059\u306d\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n \u305d\u3046\u306a\u3093\u3060\u3088\uff01\u5916\u5fc3\u3082\u4e00\u7dd2\u306b\u7406\u89e3\u3057\u3066\u304a\u3053\u3046\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n \u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u306f\u3001\u5404\u8fba\u306e\u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u3092\u66f8\u304f\u3053\u3068\u3067\u898b\u3064\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \u5fc5\u8981\u306a\u3082\u306e\u306f\u4e09\u89d2\u5f62\u3001\u7b46\u8a18\u7528\u5177\u3001\u30b3\u30f3\u30d1\u30b9<\/span>\u3067\u3059\u3002<\/p>\n\n\n\n \u307e\u305a\u3069\u3053\u304b\u306e\u8fba\u3092\u9078\u3073\u3001\u305d\u306e\u7247\u65b9\u306e\u7aef\u70b9\u304b\u3089\u30b3\u30f3\u30d1\u30b9\u3067\u5f27\u3092\u63cf\u304d\u307e\u3059\u3002<\/p>\n\n\n\n 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\u4eca\u56de\u306f\u4e94\u5fc3\u306e\u306a\u304b\u304b\u3089”\u5916\u5fc3”\u306b\u7126\u70b9\u3092\u5f53\u3066\u3066\u89e3\u8aac\u3057\u307e\u3057\u305f\u3002<\/span><\/p>\n\n\n\n \u300c\u91cd\u5fc3\u300d<\/span>\u3084\u300c\u5782\u5fc3\u300d<\/span>\u3082\u91cd\u8981\u306a\u70b9\u306a\u306e\u3067\u3001\u5fc5\u305a\u7406\u89e3\u3057\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u300c\u5916\u5fc3\u306e\u6027\u8cea\u3063\u3066\u306a\u3093\u3060\u3063\u3051\uff1f\u300d\u300c\u5916\u5fc3\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u304c\u5206\u304b\u3089\u306a\u3044\u300d\u4eca\u56de\u306f\u5916\u5fc3\u306b\u95a2\u3059\u308b\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002 \u4e09\u89d2\u5f62\u306b\u306f\u4e94\u5fc3\u3068\u547c\u3070\u308c\u308b5\u3064\u306e\u70b9\u304c\u3042\u308a\u307e\u3059\u3002 \u4eca\u56de\u89e3\u8aac\u3059\u308b\u300c\u5916\u5fc3\u300d\u306f\u4e94\u5fc3\u306e\u91cd\u8981\u306a\u70b9\u306e1\u3064\u3067\u3059\u3002 \u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u3068\u306f\u3001\u300c\u4e09 […]<\/p>\n","protected":false},"author":1,"featured_media":4920,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"swell_btn_cv_data":"","footnotes":""},"categories":[29,223],"tags":[28,10,11],"class_list":["post-1674","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-zukei","category-math-a","tag-28","tag-a","tag-11"],"yoast_head":"\n
\u300c\u5916\u5fc3\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u304c\u5206\u304b\u3089\u306a\u3044\u300d<\/span>
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\u9ad8\u6821\u751f<\/span><\/div>
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<\/figure>\n<\/div>\n\n\n\u4e09\u89d2\u5f62\u306e\u5916\u5fc3\u3068\u306f<\/h2>\n\n\n\n
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\u5404\u8fba\u306e\u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u306e\u4ea4\u70b9<\/span><\/div>\n\n\n\n
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<\/figure>\n<\/div>\n\n\n\u5916\u5fc3\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u300a\u8a3c\u660e\u300b<\/h2>\n\n\n\n
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&&\\displaystyle \\triangle PBC: \\triangle PCA: \\triangle PAB\\\\
&=&\\frac{1}{2} R^{2} \\sin 2 A: \\frac{1}{2} R^{2} \\sin 2 B: \\frac{1}{2} R^{2} \\sin 2 C\\\\
&=&\\sin 2 A: \\sin 2 B: \\sin 2 C
\\end{eqnarray}<\/p>\n\n\n\n
\u9ad8\u6821\u751f<\/span><\/div>
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\u9ad8\u6821\u751f<\/span><\/div>
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