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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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