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{ "embedding_dim" : 1024 , "data" : [ { "__id__" : "chunk-d4c4f366a47f3e13da193e0b600addae" , "__created_at__" : 1752656761 , "content" : "三角形三边关系的证明\n证明方法如下: \n作下图所示的三角形ABC。在三角形ABC中, [三角不等式](https://zhida.zhihu.com/search?content_id=248217850&content_type=Article&match_order=1&q=%E4%B8%89%E8%A7%92%E4%B8%8D%E7%AD%89%E5%BC%8F&zd_token=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpc3MiOiJ6aGlkYV9zZXJ2ZXIiLCJleHAiOjE3NTIzNzg0NDAsInEiOiLkuInop5LkuI3nrYnlvI8iLCJ6aGlkYV9zb3VyY2UiOiJlbnRpdHkiLCJjb250ZW50X2lkIjoyNDgyMTc4NTAsImNvbnRlbnRfdHlwZSI6IkFydGljbGUiLCJtYXRjaF9vcmRlciI6MSwiemRfdG9rZW4iOm51bGx9.rH6r8SvGmu-I9piEsmZfg2HjbXzUduYclZ2jfA3jZRs&zhida_source=entity)可以表示为|AB|+|BC|> |AC|。\n\nheight=\"1.90625in\"}\n①延长直线AB至点D, 并使|BD|=|BC|,连接|DC|, 那么三角形BCD为等腰三角形。所以∠BDC=∠BCD。\n②记它们均为α , 根据[欧几里得第五公理](https://zhida.zhihu.com/search?content_id=248217850&content_type=Article&match_order=1&q=%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%AC%AC%E4%BA%94%E5%85%AC%E7%90%86&zd_token=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpc3MiOiJ6aGlkYV9zZXJ2ZXIiLCJleHAiOjE3NTIzNzg0NDAsInEiOiLmrKflh6Dph4zlvpfnrKzkupTlhaznkIYiLCJ6aGlkYV9zb3VyY2UiOiJlbnRpdHkiLCJjb250ZW50X2lkIjoyNDgyMTc4NTAsImNvbnRlbnRfdHlwZSI6IkFydGljbGUiLCJtYXRjaF9vcmRlciI6MSwiemRfdG9rZW4iOm51bGx9.ltcWsMYJv-ZzcuBaSjYN69JC8hnIyPMFsfhIlum4yqc&zhida_source=entity), ∠ACD大于角∠ADC(α )。\n③由于∠ACD的对边为AD, ∠ADC(α )的对边为AC, 所以根据大角对大边([几何原本](https://zhida.zhihu.com/search?content_id=248217850&content_type=Article&match_order=1&q=%E5%87%A0%E4%BD%95%E5%8E%9F%E6%9C%AC&zd_token=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpc3MiOiJ6aGlkYV9zZXJ2ZXIiLCJleHAiOjE3NTIzNzg0NDAsInEiOiLlh6DkvZXljp_mnKwiLCJ6aGlkYV9zb3VyY2UiOiJlbnRpdHkiLCJjb250ZW50X2lkIjoyNDgyMTc4NTAsImNvbnRlbnRfdHlwZSI6IkFydGljbGUiLCJtYXRjaF9vcmRlciI6MSwiemRfdG9rZW4iOm51bGx9.Q1rCY0S2bj5Dwp3Fg7xb_VSFESz2_pCUETDybnHANvo&zhida_source=entity)中的命题19)就可以得到|AB|+|BC|=|AB|+|BD|=|AD|> |AC|。\n求证: 在三角形ABC中, P为其内部任意一点。请证明: ∠BPC > ∠A。\n证明过程: \n\n延长BP交AC于D\n∵∠BPC是△PCD的一个外角, ∠PDC是△BAD的一个外角\n∴∠BPC=∠PCD+∠PDC, ∠PDC=∠DBA+∠A\n∴∠BPC=∠PCD+∠DBA+∠A\n∴∠" , "full_doc_id" : "doc-2fa5acefbe7fd21e1f33cb1c739119ea" , "file_path" : "unknown_source" } , { "__id__" : "chunk-618195be0ad6c05ea189e352a4c2e7bb" , "__created_at__" : 1752656761 , "content" : "c35908e/media/image2.png)\n延长BP交AC于D\n∵∠BPC是△PCD的一个外角, ∠PDC是△BAD的一个外角\n∴∠BPC=∠PCD+∠PDC, ∠PDC=∠DBA+∠A\n∴∠BPC=∠PCD+∠DBA+∠A\n∴∠BPC> ∠A" , "full_doc_id" : "doc-2fa5acefbe7fd21e1f33cb1c739119ea" , "file_path" : "unknown_source" } ] , "matrix" : 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" }