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{"embedding_dim": 1024, "data": [{"__id__": "chunk-75b23a7e22383153b011bd3d121184f0", "__created_at__": 1752211461, "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![](./Images/868c80b8ff2547adb2d6b6c62c9b4c41/media/image1.png)\nheight=\"1.91044072615923in\"}\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![](./Images/868c80b8ff2547adb2d6b6c62c9b4c41/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", "full_doc_id": "doc-744d2f4e81528499ae55a82849ed415b", "file_path": "unknown_source"}, {"__id__": "chunk-e2c7bd24a26246e194d4d56ab2ed22f1", "__created_at__": 1752211461, "content": "2d6b6c62c9b4c41/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-744d2f4e81528499ae55a82849ed415b", "file_path": "unknown_source"}], "matrix": 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errxDpUy8rvqTvNy+kLwg5Yu90iNxvLKYaztzsCc7GW8MvKxKmbxw1fI6Ql7uu2oPbjyNv5M83K/6O6UeBb3EflC9LzqnPc/rqDtwAzo8F78RPXh2rD3Ow/s7MTeaPEpJEz3+FTU7X/uWvPkCuDtgF7s8sBOrvaZZ2jxzz1g8fltiPH31UrwoWwu7DJykPIvhUTxS/fQ8h68jPbde8LvqNWq8LY25vOI3nbsZbwy9VNgpPEsYv7x9qNo7F0ffvLPAGL1Vi7E7SZYLvML5D73LizM99+mguua2w7zMHwq9FVkCPQsITjzp+hS9m66fuPlPsLxOX508QvLEvP/IPD3T4oE9WFfQOz9znjy60Ii82Jz9PLETK721Jig9RnFrOm/V8rylEnw8c89YPJo5+jwrRts8Lx6DPBOphzx+W2K8iprzvJ56vjwjOfg7JBrHPFSq4jymWdo8+paOvH+RuLwOb427HZKPvZpNA720TGC8xWXIvL/KPT0LVcM87jIjPSPuSjxuQnq9gJE4PAtVQzzvz288j9geO0COiTy1wQU9KMikvPmejj3d9da7KYnVvIujaTwcXDm9/XkJPYUjSr007Ec8hTU8vLn53Dz0c/M8gL4VOxICQDyRdMq8kYa8PEupM71ub9c8g5AXPC4J9by6QSW9yy0wvexyEzxRRL67YvH5vO9fAD1F/Da9Ka25O0NOmbw/EOs8lw9VvUXYUrzoaru8naFmPLoUyLy6ZYk9F51Kuvl6KjvFd7o8PyvWvAsN+7xiS7S9ERSyO//W5TrFSl08lkUrveK0xbt6LIS8rrFVPdKtzzz0heU7NKR/u78SBrzXtIY7y0gbPeLzFL2LiH686GFCveK9PrxpHJU8S80XvZY8sjwuhxM8iDQbu4usYr3S46U8VUyWvCm2Mr1SepQ76Y4fvUUOqTwvkIw7TMQeO8sbvrmLtds7HCbjPLLdkbsX3Bm9Cw37u6kNUjwuSMS8tFVZPXm33rrSv0E8Dl0bPWkKIzkXuLW7qDqvOyUah7wpvys70uOlPDS/ajyL2T87iw8WPdKk1jyo4PQ8GskGPWkBqrxo77c51rSGvQuCIL3uOxw7BdWjPEy7JTx5wNc8bqUtPZWXjTv0ssI8o3LHu//fXjwotrI7y9zuvDF0AD3/1uW7HThVPMaJLL3XG308BXtpvaOoHTx3WoK8HAt4PCLBbbvRd3m9EbH+PPXxkbxW+rM815CivP8wID2uuk69ObSvPD2qlbyLtVu7tIuvu0gNiLux1Bg90ewePCjIJD1RKdM8vRwgO8wkN7tMsiw9JiMAPO7h4bs6xqG9xUFkPBFBDzyinyS9FQqYPIziOLxd1I08Bbo4PQXVI7xLjki9BZ/NPPlNTTuc8qe7xlNWvBykAT303x89XF9oPZdXHbzoWMk8AgwbvOmOnzxX+jO811FTvFJxGz3XfrA6RTsGPQbnlTt/f0Y8NKT/O67wpDvj4aK85deIvAx5JzxF6kQ90QcKOxamQ7gAQxI8nQ2TO5zFyryiTmO8ZkoTPBE4lryujfE8EcPwvMs/orrR0bO77HKTvAbMqrx3P5c8nR8Fva+Ncbwod2O89fGRPPlfvzwubCi8EflGvEBqJbwLVcM8gHbNO+Ki07qpDVK9HGWyPEUFMDqFCF89st2RvK+oXLyihLm8V8RdPVcDrbwLKGa9QmkEO0zfCT2WIcc7wuSHvDU9iTzivb484mx9vKKWKz2DkJc8bpO7OolGDbwgW5i8y9xuvEUXorw5h9I8d0gQPQbMqrySqiA9efatvGIeV70cJmM8VV4Ive78TL2uApe7KeOPO3nbwjzFdzo9VgMtPOKi07w8oZy8EvDNvNw0Jj3oars8+Z6OPCLcWDxcetM8tGdLvEYFsDySfcM7rEsAvGVTDD2d8ie9ZVyFO7SLLzxop+87IFsYPVHz/DtWOQO96CtsPUtzXb3psoM9S3NduvfDEzzRm928v4tuPRFBD73MWo29BWn3vFxf6LzSEAO84BgavWwJgjvvO5w9YJ2WvMltIL25LzM9Ei8dvNZ1N76oTCG8hTW8PDTjzjy1lCg80b/BOxxlMr0SHSs8NBmlvPlxMT3pfK271xt9vUbqxDwubKi8P/X/O67MwLv0u7u8HQJ/PL+C9bp5yVC8KFx4vGz3j7liFd48dH8HvLRMYLyWcog9l3uBPGjUzDt073a9eZzzO2JLtLyLBh09rEKHu+AhkzyAdk28EbH+u6OWq7z0l1c8wtuOOxxBTrx1N788NPXAPQYCgbvC0hU9tMGFPAVgfrxUQ508Y1StPNeQoryXcgg8IFKfvAW6uLzcyPk84w6Ave7Y6LxiQru7b64mPb+d4LcS51S8olfcPNqGiLyvFIm76IWmvO7GdjzjBQe9hfbsu1n5Ej2GRy48rvAkvRECQL2Mx008pXofPFxxWryo1/s3XJU+vc8+AT3uxva8S6mzPLWUqLy/efy9AEwLvcaSpbxGtG48FQGfOx1uq7z1+oq9Rav1PO8gMTysQoc9VWeBPqJFaj1ypAy9iwYdvQydCz10ZBy9esnQPJzFSry/CQ291pAivTS28byGGtE8Pz3IvOLYKbsFjdu7/99ePJKhJ73S7B49ReFLPSjIJL0cQc67uhRIvNzs3TzWkKK5AEwLvSjjD7311qa7ER0rPbnecbyuw8c6FsqnvH+Iv7vAEoY8+Rd3vQydCzzS7B697l8AvNTQErzscpO8lwbcu5frcLzS9Re8NNpVPJyzWDo/aiW9f5oxvWkclbu8QAS8ufDjPGguh7wl/xs91mw+vHRJMTx/iL+80YDyvJyY7byoBNm8V/ozPVyet7zZhgg8+Wi4PECXgjtuikI9f6MqPDb0H72RdEq9kWLYu+/q2jkrox+8X6aPPEUOKT1RXyk9uQtPvMDlqDwZrhs9L3WhvDmHUj0SJqQ8j+oQvR2JlryrJxy9aPgwvJ0fBb0cXLm74+EiPBxKx7sFcnC9KewIPcY4azzuFzi8HFy5vJWgBj0/WLM7F+4LPLodwbqCh547dCXNPPkX97ypZwy94BgavTmrNr2LKgG9ADoZvQydi7wUE5E8hf9lvR2biL0i90M9XZU+O3+IvzwoyCS9rpZqPPSg0DwCDBu8HVNAPGNULT3RyLq7zBJFPZf0aTzAEoa82X2PvLwcIL1F8708o1dcPIsGnbwX7gs9OeEMvTUHs7yx7wM8LjbSPEth6zsMnYu8P2qlvRavvDx/o6o9+qcHvblTlz0cSke84CGTPXr2Lbu5+dw8mikfvRHM6brXtIY8UTvFOgbVIz0awA28EkqIOyBSH7yd+6A8+Z6OPKOoHTzoWMk8SQQPvT9Gwby0MfU7dRPbusPSFbxjZp86Isrmuz8i3bsJ1II8yzapvCjjD7ujYNW8he1zPSy+Cr3S4yU9nLPYPMC4y7yie0C7NSsXvaPDCL3/8dA5uedqPFep8jtWDKa92XSWvNHarLoxdAA87uraPMU46zzACY07HZuIO9wHST0MZ7W7pZ6DvMaJrLxLzRc9V7JrvEX8tr1CThk+7ikqvehGV7vWbD48XJW+PGJmHz16/6Y8H2QRPXoIIL2LiP47dC7GPEkNCDz0xDQ90dosvZGGPLyAkTg9koa8OoV0C713WgK8dz8XvYz9o7zpsoO9+iBwPPloOLxANE88USBaPC4AfLwWwS48zD8ivbRMYDu5XJA7BYRivFyDTL3WfrC8DJ0LvVr5krzpjh+9XE12vBemw7yg3xS8IvfDvFHz/DxuikI8IzYTO7Z4nDxG6sQ7dEmxu8vc7ryc6S47uflcO5FH7TyXBtw8f6MqPRrADTzoK2w93ONkvXRbo7y/gnU9URdhPTJiDrxiJ1A8tpOHPW6BSTspv6s71mPFvF3dBj3oGXo716IUvVJxGz2L2T+8vwkNPaKfJDvXSFq8C1VDvFyVvrzdK608bnjQu8wtsLyRhjy9gNCHvAtntTx/SfA7v6bZvJzgNTxLoLq7qPtfOqhDqDyjV1y8o3JHvH9/Rr0oyCQ815kbPdZsvro5dWC8F5RRvMWklzx59i09v+UoPfXxEb1IFoG7P3MeuRw4Vb2ihLk8poMYvdwitLxRDmi8EUEPPWMVXrx520I9rszAu2+BybzMEsW89anJvIDZAL35g6O8OqK9vLnwY7w09cA8NUYCvBxKxzzRkmS91my+u11EfbzvX4A8yXYZuRnADbg/YSy9RaL8PLpuArzc2uu8gG3UuBUBHzySs5k8aNRMPGIM5bzL7mC74pDhvJdglry0i688qARZvChl8brjxje9uUqeu3r/przU4gQ9KfWBvLnnar0OZhQ96KkKPLWUqLzLLTC94s8wPbR5vbzuxvY86TRlvNzsXbyLvlS9TKA6PG5L8zs0GaU7BajGu1+UHTsGwzG9+rAAvdG2SL1upS09S6kzu/TEND0ptjI9HW6rvL/uobwRFDI8AgMiuxwm47yXaQ89zBLFPNG/wbypDdI8tII2va/5Hb0FcvA89swMPYVrEjwp9QG86TTlO2L68js07Ec8fPUMPbk4LL2XPDI8v8HEvOg05bxueFC8lirAvLR5vbxdsKm7op8kPBfBLj3Gtgk99Ls7vZLOhLyrMJW8gocePOt7DL711iY8rCccPJJiWLzu6lq90eyevHGSGr05kMu8Kb8rPLecgL1u0oq9gEB3vBwCfz31+oq8Lgl1PKLMATwW5ZK8F+4LPZzOQzyWKsA8tcGFO67Vubw0pP88CdSCO6PDiDmSqiC9v4vuvC9atrwdOFW8QGqlvJkpHzwFe+m8feOaPIusYj3dRpg8VhUfPQsoZrs5vSi80qTWPBEUMjzl4IE8KL8rPf/6ybyLtdu5oo2yvAyUEj3Giay8dC7GvCiA3LxFtO48nMVKOsVT1jycxco8b2bePJzFSjx/Uuk86Bn6vNwQwjv06Jg8KK25PMsA07zuFzi8P5cCvRw41bxWAy08oOgNPXKkDL0j5dG6oo2yOzT+ubyYGE68ywBTPZYqwDsR8E29LMcDO1cDrTwLH209lwbcOyIAvTyoQ6i82FpMvP/6Sb0od2O9KfWBu+Phojzvz2+8zhqdu8s2qTw0tvG87g4/vMV3OryDkBe8VvG6O//EczsoXPi8uSY6vH92TTp59i09FuUSvBELOb0OeIY8v51gPLQo/LsXwS49F4tYvUnynDzoRle8l9n+PChl8TzoNOW8YjBJvS9IRDzAABS93SstO+XFFjuLo+k8FSUDPW54UD3Jdhk9F4tYPblBpbzWqw09V7vkPOuEBb0AKKc8YlQtu4Xtcz0MlBI9nKpfvHntNLxsCQI8SRaBvaDWmzyiPHE7v8FEO1JEvj3cNKY8qOD0PCYjAL1sAIm8acvTPICROL1dlT47PxDrvJYqQDzSv0E8IG0KPcW/Aj1uVGw84mx9PEnyHDycoeY8dUC4ux2SDz05tC+97IQFvNaHqTxMoLq8r94yPKnyZjzMSJs73AfJvUtzXbz/u/q8BdUjPBHV4jzphSa8xZKlO841CDyvqNw7V9/IPABMCz3CyRy9T7oEvYUaUT0oU/87EfnGPCK49Dqc+6A9P2EsvKIzeDzL5Wc9f1LpPPR87LzshIU8Fq+8O3m3Xr2oH0Q8UjtFPEuyrLtlOKG8dEA4PIUI37uRUOa8l06kPBHV4jv5aDg8P0+6PF3dBr2RdEq8caSMvEXY0jwVChi7aBwVvYOQl7oRw3C7NAczPDlRfLzl1wi8Gq6bPDnGIT3U4gS9mjKYvT8Hcj2Ra1G8NRAsvUnynLzXqw08CbAevYlGDTy/yj09tHk9PZzXvLw/9f85IFuYu1H8dTzYP2E748a3PMD3mjyRmC69F51KvZchxzwod+M8xS/yPJo7kbtFonw8IGSRvMvu4Dw5WvU8KYlVPGZKkz2/EgY8NPVAvAtwrrw="}