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{"embedding_dim": 1024, "data": [{"__id__": "chunk-4ed981658aa2a32c616f4cf86c4efbaa", "__created_at__": 1753235662, "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![](staticImagesa6e9150381564825b2d71c00d4480801mediaimage1.png)\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![](staticImagesa6e9150381564825b2d71c00d4480801mediaimage2.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<43>", "full_doc_id": "JiHe.docx", "file_path": "JiHe.docx"}, {"__id__": "chunk-f24190139d17b1c4d499b9516ffb8a4f", "__created_at__": 1753235662, "content": "mediaimage2.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": "JiHe.docx", "file_path": "JiHe.docx"}], "matrix": 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Dhryw6w68ZVbVvEsOi7wNgIe9HFGRvA/X3TvWrLs7/vwMvGwuhbydtqM77CCauxJUczyTfYU8aAhGPH/VHL14Mxm9RGu2PWAOGzume3M8WlISPeFzrT0WBzU8G2rFPNoqIj3V4EW7weGGvOZkkboAedE8LviovVYs0zw4vfg7HFERPNXgxbvoojO8noHIPMFuCTwfAwI9TxgjPWDy87uB3mO8uACQvIETPztolhm9KnpCPMMFpLzhWFc7V/f3vD08ML08PDA8f9UcvHNdC72z2lA97K0cuue8uLz+iQ+97uwPPYxwejxLDgu9ppZJu9v1Rrz9+mo8I2XBvFHjRz0DEGw9yU4vPDKqmTzKGVS8n/UWPUlCFb36ZCE9+mShux4CsbwkMGY8WfmZPLWl9Ty+bOc8PwgmPD7KgzytxqC8wscBvbsKqDzJTd47i6XVPDuKvzwHwtw8U7COvDfu/by6gcy7OcmYvfDxCL1ngWy8uPjZvIvuXT0uVvk86pgnPaIteTwjMHy9YIIMPFZZPDzNCIQ87Y17OzP0ozw1db08AVyvvOqggD2KeiO8uACzvNfXUjw9kS69MHMKPUQcVL2K/o87YXJavDBzijzqXgo9GB+3O2cNsjv3Ori8wOKGPIYuIb2XmMc8ILe7O1jS/Ly6iSW9mZ1Nvb/iBjwgtzu8zrvhvEQU+zw9kS69d8mAO/9egrzJMLw8inJKvTM2GrysXiK9zLZbPKVHt7wJupY9A6sEu/eucjvkiZU8tHK6vEqv0rwv4r692OyKPDV1vTs16Xc8GiQ9vVx4gLzxn2C8JT1bPfL2jjxccKc7dji1u8IhKrwYo6M7vAo/PUFhHb1h7m28IH0evbhKArynwHc8MbUAvfMgejyKvBk87CEavGUAU72dt4s8dbyhvPe2S7180ww8huwqvdXapTxMrH878yD6u6M6WLzPR6c7ozrYPBokPbw3xBK9I8HHu+HGhTtBYZ28m6JTPfkFoTtxqjw8Ye7tPH/NZjuqxf05srcDPL7dgLxUngU8pcujPCZSEzzLLek6O5QBPfUt2Tx3Ax49b2sZPVBKqrz7wNe6VuWBvaeWDL1lfOY4czOvPBYKfzyG9AM9ifY2PaEDDjufvBE9m5p6u2cFWTy70KE7TAYBvYpyyjxYqBG8O4RPPIDVP724hJ88zcaNvUIPdTyBXrK8222kPFxozrvFmmq9FxfePNttpLxOyZA8rqWevCfTLD37RES9r2DVPBibyrwQ+2w7CXDHu/vA1zvfZ/48gB+PPBjlGT3Dnb08X3UtPEOomTv9USM9+TflO2qjA7xxspW9dYIEPAtDCTwaqCm9rWOoPLNq4bu0cro8M3A3PRqgULwSECW9Llb5PAVeYjt8yzM7e8szvNObAj03Bgk9Ll5SPT+WNLvC54w8m6LTu61jqDwz5HG8OxCVvEy8MT1kjBi6zYQXPVRM3Tvq2h08Eh0EPODzQzxf6We8kxKovDV9ljt1rG89jggcPEGburoGY2g8FIllO0Qkrbz3y4O8rFZJPPMo07yybbQ8kwrPvLqBTDurJAW7Wt9bvJnnnLwjwUc89zLfvAMnGLu4dG282S6BPAfvrTyL7l28HHMSvGV85rsz7Mo8PQVpPFioETuVD1W9yaT2PNnksbuxZVs9czMvvP/SPLzrGcG8OXdwPTHfa7xnDTK9exWDO9ttJD2vWPw7Owi8vHGqvDxgerM8Zfh5vHrGLT3iAKM82eSxujt8drtrH5e8aNiavMLnjLzgb1c8gV4yPZUHfLxrowM9hN/LvCO0aL0CWdw7RTEMvcMZUb3FFn66O1ILPF3sujwx50Q9ikAGPE7Bt7ybLpm8TECevBCHMj2PfNY8vReePHvDWjyXmMc8MedEujExlDz4ywM7GZh3vNUcHD24xhW99bFFPP1RIzyUH4e6Dv4/PSrYMjuybbS8cTaCPQPdSL2Q04Q9Uk8wu8m0qDt6vtS87Y17PdVWOb3bM4e9Sn0OvXAZ8bwATH28a5sqvfs86zu09qY9FFehvEmn+bxuIUo9MWPYu9zpN74Yo6O76A+1PHEuqTxf6ec74OvqOtESEL3ealE8tLwJvLoFOT28jis7X2V7vf2TmTyI8bC8iv6Pu21mE7zzKFO8GiS9PP0XhjpBHye8Vh+fvD4F6ToHtZA8hN9LvDFbf7xukH49LVFzPGPzczwD1W+9DgufO8Wa6ryuYyg9kQVJu4zBnzyYnU28TsmQvODr6ry/i9g7m3APPAVuFLyqSeo8RaXGPVRcD7oqnhU9TLRYPBKMuLvrEWg8Md/rPMEUS7yWmMc6IThVvNE8+7xqD+U8mRlhvbsSGL1b39s5SrcrPbS8CTxtoLC7dCvWPDPsyrxdYHW8bymjvFLLQzzffwm9UhWTuX6WHD1ORSQ8SrcrvTX5Kb10K9Y7OH/JurZvZ7xUnoU6KFcZvZ0j7TxgxAK95hKIPJWTwbzON/W9I48DvWbTlLy2A4Y8KuCLO99v17wdNqK9WF7CPPI4hTyowHc9t0qCPlnaVT3/HAy99S3ZvEsw7DyKeiO96FmEPP1Ro7zfPRO9uMYVveJCGb0Ocvo80ETUvIpySjsesjW7/UnKO82+NL3+Vik9IDNPPZckDb1CF8668K8SvCvYsjzRPPu7nXUVve8W7rwsYSW8EgDzPFTYorvV2iW6UIygvD2ZB7tOyZA8wOKGvSSPgztOCwe9tvNTu6zaNbz1sUW8AVyvuzM2GrwD5SG8x6dJPFLLQzs3ghy9198rvc+7YbsWjmu8huTRPITfS7z0PQs9qySFvIvuXTwFZru8/RcGvT8K77wcnf28iPEwPfc6uLzPR6c8joSvPKjI0Ds/GiE9yXqLOyaUCb1lfGa9qVmcuzV1vbujQjG8TkWkPEQkLT1WWTw97xbuu+UKrzwB4Bs9HrK1vIDNZj1MxIo8IK/ivGUA07xQSiq9mCE6vMCgEL3jR5+6czMvPAk+A7w+BWm9i+7dPEwGgTx+2BK8ILe7vDJo3jw3cuo7WNrVOzM2GjtaMQQ8SKrMPNtd8rxpXAe90xcWvWeJRb1aW++8TgsHvZSLaLy4SoI8QBJIvVtbb71ILjk9sHAHvDfu/TwaqCm9WltvPD+WtDyOjIi7MzaaPHGqPD3YU2a7DnpTPfMoUzxBmzq89PuUvLgIDL0bKcM8Rp1tPC9mq7xx9As9HroOveRHn7xjA6Y8DPHgPHc1Yjw7fHa8IsSavWFAljzok6E99PsUvb7dgD3zKFO8hnCXPSiZD7tMQJ48Kp4VvS1ZTDlOyZA7CnXNOhaWRD2I6de7tLwJPFfN9rtngew8GiQ9PBB/WTwgQwE9Zfj5vEOgwLyCXjI7svGgu4r2Nrx8yzO74G9Xuxijo7vvI0086hyUvMQuiTmf/oe8mZV0PXvD2rwLvxw9pcujPNPF7byE5yQ6M3gQvct/Eb30fwG7sm00PIDNZjoDYbW90UytvGmWJLvmAlY8Z4HsPKM62Dx+2JI7kQVJPHSvQj3iACO8d7F1vCZKurz4Rxc9YHozvK9orr1pGhE+huwqvT8SyLtYVuk7COzaPNFMLT2cM588220kPTN4EL0vXtI7i+7dPAXqJzw1+Sk9bagJvcMR+Lu2d0A9waWWOiI41by/i1i8VuUBvVTI8LzaWGy9XLKdPI50fbyaJsA7VNiiO/cyX7wiONU7PQ1CvQt9JjtOud47nnobu2J/Ob3AmLe8eTrovA72Zrw8FRu9zz9OuzHnxLwmUhO8FiIKvcAcJD2bolM8uHRtO867YTzsIZo7O84eOm8pI73BDPI75Hlju2cNsjwe/IQ8YMQCPStURjw7CDw9HSZwvbJl27sF2nU9YXJaPXFwH7zc4V48hFtfPSB9njsWjms6Ry65vC1hJT2sXiI7OQM2vU5FJD2rJIW8QeUJPdjsCjw9DUK8vQJmuzN4kLzd9pY8jPs8vIr+j7yMfym9WjGEvHAZ8TxHLjk8JbnuvNttJDw9kS68bagJu9hTZjxaY0i8kYk1vIbkUb3Nxo08wp09PeoUO7uE13K8miZAvKcSIDwH51Q9xys2PRqoKb14wac4EZQROopySr0x75085n7pvHpCwbwHtRC8huwqPQn8jLzX3ys9uYklvPuOE73Pw7q819fSvFVZvLxDLIa8PZGuvKcSoLzeYvg8nK8yvHyRljysTnC9MN/rutHAZ7w1v4w8QeWJu2f9/7mZITq9XOThPNXSzLur2rW8v6CQuqeWDDxUyHA8y7FVO8CQ3rx+lpy5wJDevBpujLweLsk8r2DVuzFrMbzmhkK9cbIVvDkLj7zVXhI9ZAgsvHIed735wyo98yD6O0ii87ybolO9zj9OPcs9m7xUyPA8Zw2yvD1Xkbzg80O9WVZpO4TX8js9gXw8AdDpu4Afj7uJ9ja9IAELvc+7Yb2QkQ499YSHO7wKPz0gtzs9MTGUvNVWubxSV4k8GiwWvCjbBb1ohvI819fSPDnJmLzAkN484vhJvSpcH73w8Qg92zMHPRibSjwRlJG8IL8UPCQLFzzxG3Q8iDsAPb6TMb1Qxr07uv3fvM/DurxHLjm8uISfvGthDb3qmCe7iDuAPIp6Iz30PQs99gAbvQp1TbxEFHu8OfvcO3eHCr7torM7qs3WO0adbbxSw2q9K9iyvAzx4Lx4uc68UEoqPFnvjb3u7IK9hOekvK9whz205nS8GiS9PGgK3zpZVum8zk8APStMbTzGdQU9myZAOuoUu7xLrP88vI6rOtttpDucMx+9yy3pvHxPoLw7hE+8O1KLvKMIlDwNgqy8xp9wPP0Xhj3mEgg8We8NPQttdLulyyO8222kPFCMoDxOC4c8oX8hPQpo7rxzO4i75HnjvJMSKD29F568xyPdvJmV9LzAoBA9OwBjO84/zjzHr6I8cTYCPXg9OzwxY9g8FhLYvFoxhDvecqo8qVHDPHsVA73HI127bagJvS/a5bx6xq08B/cGPR66Dr25yxs8B7WQOyK8wbzELom8EH9ZPUIPdTyZGWG9er5UusSqnDz7wFc9gEl6O88NCjx8kZa8dYIEvJSLaL3LqXy9i0WMuG1mkzyGYGW8rNJcu/ZCkTx9TM28FA1SvBaenbx7w1q84gAjPBytLzzcZcu8TsmQvI4InLsJuhY9Sivmux02Ir2iLXk8y38RPGuT0btU2CI98qRmvaQyfzw5+9y8jowIPSlH5zy07s28hmg+vX9ZLDyAHw+9TsG3O9NJ2jt8y7M85hIIPdpgRT1m/f88FpZEPdlonrxTP/48/9K8PC8sDr19VKY8zDrIu/1JSj2J4f48AEx9vCbOJrwD3Ug8mWuJvQm6ljz7uH67zw2KO48Awz0aHOQ7seHuPOT19rwJ/Iy8jW/3POwhGr2AYYW6sNzovChXGTy1d0A8A2kOPct/ET3PiZ08vJaEPGhchzvuorM83mJ4vMEUSz1CF0693nKqu1qtlzzoD7W8iPGwO1A6eDwSjLg6pUe3vZ+8kbxKfQ69A6uEO/mz+DxBHye8LdXfO//aFTzJrM87gGGFPMQuCT27Ehi9xZrqvAn0Mz24SoI8P5a0PCABCzyXoKA9kpaUu3eHCjwqIoI9DYoFPXWs77wNBpk8zDpIPOiDb70bIWo8HLWIPOyV1Lu2+6y8ZxWLPDf+L7w7fPa8UMa9POjVFzxrk1E8gFHTPCwnCL1NCwe8ERAlvBQVqzyn2AK8yLwBvd1ieLueOCW8FpbEPNFMLby0vIm8+QWhPPArJj2XJA29AyeYvQPVbz3+Rne8NfkpvYLinrxc9JM7LaMbvS9mKzzJrE89+cMqPeqYp7y9D0W5BV5iuh66jjuyZds7O4yoPD9clzybqiy9sm00vTsIvDzxn+A8tPamPFrfW7zc4V48MWPYvBRXoTyC0uw8Xn2GPMk4lT3yMKw54nRdvKxWybw="}