消元难题

数学星空

经过好长时间的计算，最终我们得到了需要的计算结果：


\(bc(-c+a-b)(a^3+3a^2b+3a^2c-ab^2+6abc-ac^2-3b^3-9b^2c-9bc^2-3c^3)(a^4-2a^3b-2a^3c-7a^2b^2-6a^2bc-7a^2c^2+2ab^3+22ab^2c+22abc^2+2ac^3+6b^4+36b^3c+60b^2c^2+36bc^3+6c^4)+2(a+b+c)(-c+a-b)(2a^6b+2a^6c-4a^5b^2+26a^5bc-4a^5c^2-16a^4b^3-31a^4b^2c-31a^4bc^2-16a^4c^3+8a^3b^4-122a^3b^3c-204a^3b^2c^2-122a^3bc^3+8a^3c^4+26a^2b^5+58a^2b^4c+24a^2b^3c^2+24a^2b^2c^3+58a^2bc^4+26a^2c^5-4ab^6+112ab^5c+332ab^4c^2+272ab^3c^3+332ab^2c^4+112abc^5-4ac^6-12b^7-45b^6c-117b^5c^2-258b^4c^3-258b^3c^4-117b^2c^5-45bc^6-12c^7)x+(16a^8+80a^7b+80a^7c-40a^6b^2+241a^6bc-40a^6c^2-368a^5b^3-398a^5b^2c-398a^5bc^2-368a^5c^3-40a^4b^4-697a^4b^3c-710a^4b^2c^2-697a^4bc^3-40a^4c^4+496a^3b^5+908a^3b^4c+1748a^3b^3c^2+1748a^3b^2c^3+908a^3bc^4+496a^3c^5+136a^2b^6+895a^2b^5c+2796a^2b^4c^2+4538a^2b^3c^3+2796a^2b^2c^4+895a^2bc^5+136a^2c^6-208ab^7-462ab^6c-966ab^5c^2-2108ab^4c^3-2108ab^3c^4-966ab^2c^5-462abc^6-208ac^7-72b^8-567b^7c-2430b^6c^2-5913b^5c^3-7956b^4c^4-5913b^3c^5-2430b^2c^6-567bc^7-72c^8)x^2+4(a+b+c)(16a^6+45a^5b+45a^5c-29a^4b^2-168a^4bc-29a^4c^2-202a^3b^3-494a^3b^2c-494a^3bc^2-202a^3c^3-6a^2b^4+332a^2b^3c-292a^2b^2c^2+332a^2bc^3-6a^2c^4+221ab^5+561ab^4c+1234ab^3c^2+1234ab^2c^3+561abc^4+221ac^5-45b^6-276b^5c-579b^4c^2-792b^3c^3-579b^2c^4-276bc^5-45c^6)x^3+(656a^6+1344a^5b+1344a^5c-368a^4b^2+1900a^4bc-368a^4c^2-1664a^3b^3-2528a^3b^2c-2528a^3bc^2-1664a^3c^3-720a^2b^4-3080a^2b^3c-800a^2b^2c^2-3080a^2bc^3-720a^2c^4+320ab^5+3072ab^4c+7552ab^3c^2+7552ab^2c^3+3072abc^4+320ac^5+432b^6+3132b^5c+8496b^4c^2+12168b^3c^3+8496b^2c^4+3132bc^5+432c^6)x^4-96(a+b+c)(4a^4-14a^3b-14a^3c-10a^2b^2+50a^2bc-10a^2c^2+28ab^3+41ab^2c+41abc^2+28ac^3-8b^4-37b^3c-18b^2c^2-37bc^3-8c^4)x^5+(576a^4-144a^2bc-6048ab^2c-6048abc^2-576b^4-7056b^3c-7200b^2c^2-7056bc^3-576c^4)x^6+1728(a+b+c)(ab+ac-b^2-c^2)x^7+5184bcx^8＝0\)

1. ca(-a+b-c)(-3a^3-a^2b-9a^2c+3ab^2+6abc-9ac^2+b^3+3b^2c-bc^2-3c^3)(6a^4+2a^3b+36a^3c-7a^2b^2+22a^2bc+60a^2c^2-2ab^3-6ab^2c+22abc^2+36ac^3+b^4-2b^3c-7b^2c^2+2bc^3+6c^4)+2(a+b+c)(-a+b-c)(-12a^7-4a^6b-45a^6c+26a^5b^2+112a^5bc-117a^5c^2+8a^4b^3+58a^4b^2c+332a^4bc^2-258a^4c^3-16a^3b^4-122a^3b^3c+24a^3b^2c^2+272a^3bc^3-258a^3c^4-4a^2b^5-31a^2b^4c-204a^2b^3c^2+24a^2b^2c^3+332a^2bc^4-117a^2c^5+2ab^6+26ab^5c-31ab^4c^2-122ab^3c^3+58ab^2c^4+112abc^5-45ac^6+2b^6c-4b^5c^2-16b^4c^3+8b^3c^4+26b^2c^5-4bc^6-12c^7)y+(-72a^8-208a^7b-567a^7c+136a^6b^2-462a^6bc-2430a^6c^2+496a^5b^3+895a^5b^2c-966a^5bc^2-5913a^5c^3-40a^4b^4+908a^4b^3c+2796a^4b^2c^2-2108a^4bc^3-7956a^4c^4-368a^3b^5-697a^3b^4c+1748a^3b^3c^2+4538a^3b^2c^3-2108a^3bc^4-5913a^3c^5-40a^2b^6-398a^2b^5c-710a^2b^4c^2+1748a^2b^3c^3+2796a^2b^2c^4-966a^2bc^5-2430a^2c^6+80ab^7+241ab^6c-398ab^5c^2-697ab^4c^3+908ab^3c^4+895ab^2c^5-462abc^6-567ac^7+16b^8+80b^7c-40b^6c^2-368b^5c^3-40b^4c^4+496b^3c^5+136b^2c^6-208bc^7-72c^8)y^2+4(a+b+c)(-45a^6+221a^5b-276a^5c-6a^4b^2+561a^4bc-579a^4c^2-202a^3b^3+332a^3b^2c+1234a^3bc^2-792a^3c^3-29a^2b^4-494a^2b^3c-292a^2b^2c^2+1234a^2bc^3-579a^2c^4+45ab^5-168ab^4c-494ab^3c^2+332ab^2c^3+561abc^4-276ac^5+16b^6+45b^5c-29b^4c^2-202b^3c^3-6b^2c^4+221bc^5-45c^6)y^3+(432a^6+320a^5b+3132a^5c-720a^4b^2+3072a^4bc+8496a^4c^2-1664a^3b^3-3080a^3b^2c+7552a^3bc^2+12168a^3c^3-368a^2b^4-2528a^2b^3c-800a^2b^2c^2+7552a^2bc^3+8496a^2c^4+1344ab^5+1900ab^4c-2528ab^3c^2-3080ab^2c^3+3072abc^4+3132ac^5+656b^6+1344b^5c-368b^4c^2-1664b^3c^3-720b^2c^4+320bc^5+432c^6)y^4-96(a+b+c)(-8a^4+28a^3b-37a^3c-10a^2b^2+41a^2bc-18a^2c^2-14ab^3+50ab^2c+41abc^2-37ac^3+4b^4-14b^3c-10b^2c^2+28bc^3-8c^4)y^5+(-576a^4-7056a^3c-6048a^2bc-7200a^2c^2-144ab^2c-6048abc^2-7056ac^3+576b^4-576c^4)y^6+1728(a+b+c)(-a^2+ab+bc-c^2)y^7+5184cay^8＝0

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1. ab(-b+c-a)(-3a^3-9a^2b-a^2c-9ab^2+6abc+3ac^2-3b^3-b^2c+3bc^2+c^3)(6a^4+36a^3b+2a^3c+60a^2b^2+22a^2bc-7a^2c^2+36ab^3+22ab^2c-6abc^2-2ac^3+6b^4+2b^3c-7b^2c^2-2bc^3+c^4)+2(a+b+c)(-b+c-a)(-12a^7-45a^6b-4a^6c-117a^5b^2+112a^5bc+26a^5c^2-258a^4b^3+332a^4b^2c+58a^4bc^2+8a^4c^3-258a^3b^4+272a^3b^3c+24a^3b^2c^2-122a^3bc^3-16a^3c^4-117a^2b^5+332a^2b^4c+24a^2b^3c^2-204a^2b^2c^3-31a^2bc^4-4a^2c^5-45ab^6+112ab^5c+58ab^4c^2-122ab^3c^3-31ab^2c^4+26abc^5+2ac^6-12b^7-4b^6c+26b^5c^2+8b^4c^3-16b^3c^4-4b^2c^5+2bc^6)z+(-72a^8-567a^7b-208a^7c-2430a^6b^2-462a^6bc+136a^6c^2-5913a^5b^3-966a^5b^2c+895a^5bc^2+496a^5c^3-7956a^4b^4-2108a^4b^3c+2796a^4b^2c^2+908a^4bc^3-40a^4c^4-5913a^3b^5-2108a^3b^4c+4538a^3b^3c^2+1748a^3b^2c^3-697a^3bc^4-368a^3c^5-2430a^2b^6-966a^2b^5c+2796a^2b^4c^2+1748a^2b^3c^3-710a^2b^2c^4-398a^2bc^5-40a^2c^6-567ab^7-462ab^6c+895ab^5c^2+908ab^4c^3-697ab^3c^4-398ab^2c^5+241abc^6+80ac^7-72b^8-208b^7c+136b^6c^2+496b^5c^3-40b^4c^4-368b^3c^5-40b^2c^6+80bc^7+16c^8)z^2+4(a+b+c)(-45a^6-276a^5b+221a^5c-579a^4b^2+561a^4bc-6a^4c^2-792a^3b^3+1234a^3b^2c+332a^3bc^2-202a^3c^3-579a^2b^4+1234a^2b^3c-292a^2b^2c^2-494a^2bc^3-29a^2c^4-276ab^5+561ab^4c+332ab^3c^2-494ab^2c^3-168abc^4+45ac^5-45b^6+221b^5c-6b^4c^2-202b^3c^3-29b^2c^4+45bc^5+16c^6)z^3+(432a^6+3132a^5b+320a^5c+8496a^4b^2+3072a^4bc-720a^4c^2+12168a^3b^3+7552a^3b^2c-3080a^3bc^2-1664a^3c^3+8496a^2b^4+7552a^2b^3c-800a^2b^2c^2-2528a^2bc^3-368a^2c^4+3132ab^5+3072ab^4c-3080ab^3c^2-2528ab^2c^3+1900abc^4+1344ac^5+432b^6+320b^5c-720b^4c^2-1664b^3c^3-368b^2c^4+1344bc^5+656c^6)z^4-96(a+b+c)(-8a^4-37a^3b+28a^3c-18a^2b^2+41a^2bc-10a^2c^2-37ab^3+41ab^2c+50abc^2-14ac^3-8b^4+28b^3c-10b^2c^2-14bc^3+4c^4)z^5+(-576a^4-7056a^3b-7200a^2b^2-6048a^2bc-7056ab^3-6048ab^2c-144abc^2-576b^4+576c^4)z^6+1728(a+b+c)(-a^2+ac-b^2+bc)z^7+5184abz^8＝0

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数学星空

若要求此内点\(P\)到三边\(a,b,c\)的距离\(h_1,h_2,h_3\) (即楼上提到未解决的三线坐标表示）关于\(a,b,c\)的代数方程？
谁有兴趣破解这个难题(即8#提到的X(5394) = CONGRUENT INCIRCLES POINT)！！

数学星空

若记\(s_1=S_{BPC},s_2=S_{APC},s_3=S_{APB}\)

1. 16a^4(3a^3+9a^2b+a^2c+9ab^2-6abc-3ac^2+3b^3+b^2c-3bc^2-c^3)(3a^3+a^2b+9a^2c-3ab^2-6abc+9ac^2-b^3-3b^2c+bc^2+3c^3)-16a(42a^8+151a^7b+151a^7c+191a^6b^2-178a^6bc+191a^6c^2+23a^5b^3-643a^5b^2c-643a^5bc^2+23a^5c^3-173a^4b^4-268a^4b^3c+482a^4b^2c^2-268a^4bc^3-173a^4c^4-163a^3b^5-63a^3b^4c+226a^3b^3c^2+226a^3b^2c^3-63a^3bc^4-163a^3c^5-59a^2b^6+30a^2b^5c+59a^2b^4c^2-60a^2b^3c^3+59a^2b^2c^4+30a^2bc^5-59a^2c^6-11ab^7+11ab^6c+33ab^5c^2-33ab^4c^3-33ab^3c^4+33ab^2c^5+11abc^6-11ac^7-b^8+4b^6c^2-6b^4c^4+4b^2c^6-c^8)t_1+(2068a^8+7888a^7b+7888a^7c+8848a^6b^2-23360a^6bc+8848a^6c^2+528a^5b^3-75472a^5b^2c-75472a^5bc^2+528a^5c^3-8296a^4b^4-19712a^4b^3c+10256a^4b^2c^2-19712a^4bc^3-8296a^4c^4-8080a^3b^5+688a^3b^4c+7392a^3b^3c^2+7392a^3b^2c^3+688a^3bc^4-8080a^3c^5-2608a^2b^6+2112a^2b^5c+2608a^2b^4c^2-4224a^2b^3c^3+2608a^2b^2c^4+2112a^2bc^5-2608a^2c^6-336ab^7+336ab^6c+1008ab^5c^2-1008ab^4c^3-1008ab^3c^4+1008ab^2c^5+336abc^6-336ac^7-12b^8+48b^6c^2-72b^4c^4+48b^2c^6-12c^8)t_1^2+(-5456a^7-25936a^6b-25936a^6c-15056a^5b^2+5088a^5bc-15056a^5c^2+14384a^4b^3+217104a^4b^2c+217104a^4bc^2+14384a^4c^3+18064a^3b^4+53568a^3b^3c-16288a^3b^2c^2+53568a^3bc^3+18064a^3c^4+11408a^2b^5+976a^2b^4c-12384a^2b^3c^2-12384a^2b^2c^3+976a^2bc^4+11408a^2c^5+2448ab^6-1824ab^5c-2448ab^4c^2+3648ab^3c^3-2448ab^2c^4-1824abc^5+2448ac^6+144b^7-144b^6c-432b^5c^2+432b^4c^3+432b^3c^4-432b^2c^5-144bc^6+144c^7)t_1^3+(8912a^6+59904a^5b+59904a^5c+16912a^4b^2+200608a^4bc+16912a^4c^2-52352a^3b^3-321536a^3b^2c-321536a^3bc^2-52352a^3c^3-25104a^2b^4-78272a^2b^3c+36256a^2b^2c^2-78272a^2bc^3-25104a^2c^4-7552ab^5-5376ab^4c+12928ab^3c^2+12928ab^2c^3-5376abc^4-7552ac^5-720b^6+288b^5c+720b^4c^2-576b^3c^3+720b^2c^4+288bc^5-720c^6)t_1^4+(-7296a^5-77952a^4b-77952a^4c-11520a^3b^2-536064a^3bc-11520a^3c^2+76032a^2b^3+255744a^2b^2c+255744a^2bc^2+76032a^2c^3+18816ab^4+56832ab^3c-40704ab^2c^2+56832abc^3+18816ac^4+1920b^5+3456b^4c-5376b^3c^2-5376b^2c^3+3456bc^4+1920c^5)t_1^5+(2304a^4+51840a^3b+51840a^3c+3456a^2b^2+626688a^2bc+3456a^2c^2-51840ab^3-100224ab^2c-100224abc^2-51840ac^3-5760b^4-16128b^3c+16128b^2c^2-16128bc^3-5760c^4)t_1^6+(-13824a^2b-13824a^2c-359424abc+13824b^3+13824b^2c+13824bc^2+13824c^3)t_1^7+82944cbt_1^8=0

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分别将上式变量作代换：

\(a=b,b=c,c=a,t_1=t_2\)

\(a=c,b=a,c=b,t_1=t_3\)

则可以得到：\(s_1:s_2:s_3=r(t_1:t_2:t_3)\)

1. 324a^6bck_1^8+144a^8+2208a^4b^2c^2-576a^4bc^3-672a^3b^4c+960a^3b^3c^2+960a^3b^2c^3-672a^3bc^4-160a^2b^5c+48a^2b^4c^2+320a^2b^3c^3+48a^2b^2c^4-160a^2bc^5+736a^6bc+
   
2. 192a^5b^2c+192a^5bc^2-576a^4b^3c+(-108a^7b-108a^7c-2808a^6bc+108a^5b^3+108a^5b^2c+108a^5bc^2+108a^5c^3)k_1^7+(36a^8+810a^7b+810a^7c+54a^6b^2+9792a^6bc+54a^6c^2-810a^5b^3-1566a^5b^2c-1566a^5bc^2-810a^5c^3-90a^4b^4-252a^4b^3c+252a^4b^2c^2-252a^4bc^3-90a^4c^4)k_1^6+(-228a^8-2436a^7b-2436a^7c-360a^6b^2-16752a^6bc-360a^6c^2+2376a^5b^3+7992a^5b^2c+7992a^5bc^2+2376a^5c^3+588a^4b^4+1776a^4b^3c-1272a^4b^2c^2+1776a^4bc^3+588a^4c^4+60a^3b^5+108a^3b^4c-168a^3b^3c^2-168a^3b^2c^3+108a^3bc^4+60a^3c^5)k_1^5+(557a^8+3744a^7b+3744a^7c+1057a^6b^2+12538a^6bc+1057a^6c^2-3272a^5b^3-20096a^5b^2c-20096a^5bc^2-3272a^5c^3-1569a^4b^4-4892a^4b^3c+2266a^4b^2c^2-4892a^4bc^3-1569a^4c^4-472a^3b^5-336a^3b^4c+808a^3b^3c^2+808a^3b^2c^3-336a^3bc^4-472a^3c^5-45a^2b^6+18a^2b^5c+45a^2b^4c^2-36a^2b^3c^3+45a^2b^2c^4+18a^2bc^5-45a^2c^6)k_1^4+(-682a^8-3242a^7b-3242a^7c-1882a^6b^2+636a^6bc-1882a^6c^2+1798a^5b^3+27138a^5b^2c+27138a^5bc^2+1798a^5c^3+2258a^4b^4+6696a^4b^3c-2036a^4b^2c^2+6696a^4bc^3+2258a^4c^4+1426a^3b^5+122a^3b^4c-1548a^3b^3c^2-1548a^3b^2c^3+122a^3bc^4+1426a^3c^5+306a^2b^6-228a^2b^5c-306a^2b^4c^2+456a^2b^3c^3-306a^2b^2c^4-228a^2bc^5+306a^2c^6+18ab^7-18ab^6c-54ab^5c^2+54ab^4c^3+54ab^3c^4-54ab^2c^5-18abc^6+18ac^7)k_1^3+(517a^8+1972a^7b+1972a^7c+2212a^6b^2-5840a^6bc+2212a^6c^2+132a^5b^3-18868a^5b^2c-18868a^5bc^2+132a^5c^3-2074a^4b^4-4928a^4b^3c+2564a^4b^2c^2-4928a^4bc^3-2074a^4c^4-2020a^3b^5+172a^3b^4c+1848a^3b^3c^2+1848a^3b^2c^3+172a^3bc^4-2020a^3c^5-652a^2b^6+528a^2b^5c+652a^2b^4c^2-1056a^2b^3c^3+652a^2b^2c^4+528a^2bc^5-652a^2c^6-84ab^7+84ab^6c+252ab^5c^2-252ab^4c^3-252ab^3c^4+252ab^2c^5+84abc^6-84ac^7-3b^8+12b^6c^2-18b^4c^4+12b^2c^6-3c^8)k_1^2+(-336a^8-1208a^7b-1208a^7c-1528a^6b^2+1424a^6bc-1528a^6c^2-184a^5b^3+5144a^5b^2c+5144a^5bc^2-184a^5c^3+1384a^4b^4+2144a^4b^3c-3856a^4b^2c^2+2144a^4bc^3+1384a^4c^4+1304a^3b^5+504a^3b^4c-1808a^3b^3c^2-1808a^3b^2c^3+504a^3bc^4+1304a^3c^5+472a^2b^6-240a^2b^5c-472a^2b^4c^2+480a^2b^3c^3-472a^2b^2c^4-240a^2bc^5+472a^2c^6+88ab^7-88ab^6c-264ab^5c^2+264ab^4c^3+264ab^3c^4-264ab^2c^5-88abc^6+88ac^7+8b^8-32b^6c^2+48b^4c^4-32b^2c^6+8c^8)k_1+432a^6c^2+432a^6b^2-192a^5b^3-192a^5c^3+480a^7b+480a^7c-528a^4b^4-528a^4c^4-288a^3b^5-288a^3c^5-48a^2b^6-48a^2c^6=0

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分别将上式变量作代换：

\(a=b,b=c,c=a,k_1=k_2\)

\(a=c,b=a,c=b,k_1=k_3\)

则可以得到：\(h_1:h_2:h_3=r(k_1:k_2:k_3)\)