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# [提问] 三角形内三个四边形的内切圆

 为了方便计算，我们可以得到更简洁的形式，大曲边三角形的代数曲线： \(-2a(a+b-c)(a-b+c)(-b-c+a)(2a^3-2a^2b+2a^2c-ab^2-2abc+3ac^2+b^3-b^2c-bc^2+c^3)xy^2+(2(a+b-c))(a-b+c)(-b-c+a)(2a^3+ab^2-2abc+ac^2+b^3-b^2c-bc^2+c^3)x^2y^2+a^2(c+a+b)(-b-c+a)(a+b-c)(a-b+c)^3y^2+(2a^3-ab^2+2abc-ac^2-b^3+b^2c+bc^2-c^3)^2y^4+a^2(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^2-2a(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^3+(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^4=0\) \(-4a(a+b-c)(a-b+c)(-b-c+a)(a^5+3a^4b-3a^4c-2a^3b^2-8a^3bc+2a^3c^2+2a^2b^3-10a^2b^2c+6a^2bc^2+2a^2c^3-3ab^4+6ab^2c^2-3ac^4-b^5+b^4c+2b^3c^2-2b^2c^3-bc^4+c^5)(a^2-2ab-2ac-b^2+c^2)xy^2+14a^{12}bc+(a^5+3a^4b-3a^4c+6a^3b^2-8a^3bc+2a^3c^2-6a^2b^3-2a^2b^2c+6a^2bc^2+2a^2c^3-3ab^4+6ab^2c^2-3ac^4-b^5+b^4c+2b^3c^2-2b^2c^3-bc^4+c^5)^2y^4-6a^{13}c-40a^{10}b^3c-2a^{13}b+(2(a+b-c))(a-b+c)(-b-c+a)(a^7-a^6b-5a^6c-7a^5b^2-2a^5bc+9a^5c^2+31a^4b^3+21a^4b^2c+17a^4bc^2-5a^4c^3+15a^3b^4+28a^3b^3c-10a^3b^2c^2-28a^3bc^3-5a^3c^4+17a^2b^5+9a^2b^4c-34a^2b^3c^2-18a^2b^2c^3+17a^2bc^4+9a^2c^5+7ab^6-2ab^5c-19ab^4c^2+4ab^3c^3+17ab^2c^4-2abc^5-5ac^6+b^7-b^6c-3b^5c^2+3b^4c^3+3b^3c^4-3b^2c^5-bc^6+c^7)x^2y^2+(a-b+c)^2(-b-c+a)^2(a+b-c)^2(a^2-4ab-2ac-b^2+c^2)^2x^4+12a^5b^4c^5+36a^5b^5c^4-2a^4b^9c-40a^6bc^7+12a^6b^2c^6+72a^6b^3c^5-66a^6b^4c^4-24a^5b^6c^3-8a^5b^7c^2-4a(a^2-2ab-2ac-b^2+c^2)(a^2-4ab-2ac-b^2+c^2)(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^3+110a^8b^2c^4-24a^6b^5c^3+13a^{12}c^2-3a^{12}b^2+10a^5b^8c+56a^7bc^6+104a^7b^4c^3-24a^7b^5c^2-24a^7b^6c-72a^9b^2c^3+44a^6b^6c^2-8a^6b^7c-28a^8bc^5+72a^9b^3c^2-72a^7b^2c^5-40a^7b^3c^4+36a^8b^5c+56a^{10}bc^3+12a^{10}b^2c^2+2a^2(a-b+c)(a^5+3a^4b-3a^4c-2a^3b^2-8a^3bc+2a^3c^2+2a^2b^3-10a^2b^2c+6a^2bc^2+2a^2c^3-3ab^4+6ab^2c^2-3ac^4-b^5+b^4c+2b^3c^2-2b^2c^3-bc^4+c^5)(-b-c+a)^2(a+b-c)^2y^2-4a^3(a^2-2ab-2ac-b^2+c^2)(a-b+c)^2(-b-c+a)^3(a+b-c)^3x+2a^2(3a^4-12a^3b-12a^3c+2a^2b^2+24a^2bc+18a^2c^2+12ab^3+12ab^2c-12abc^2-12ac^3+3b^4-6b^2c^2+3c^4)(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^2-40a^8b^3c^3-66a^8b^4c^2+a^{14}+8a^5b^2c^7-28a^9bc^4+12a^9b^4c-6a^5c^9-2a^5b^9-8a^7c^7-40a^{11}bc^2+8a^{11}b^2c-2a^4bc^9-3a^4b^2c^8+8a^7b^7+28a^9c^5-12a^9b^5-8a^{11}c^3+8a^{11}b^3+a^4c^{10}+a^4b^{10}+13a^6c^8-3a^6b^8+14a^5bc^8-40a^5b^3c^6-14a^8c^6+2a^8b^6+8a^4b^3c^7+2a^4b^4c^6-12a^4b^5c^5-14a^{10}c^4+2a^{10}b^4+2a^4b^6c^4+8a^4b^7c^3-3a^4b^8c^2=0\) \(8a^2c(a+b-c)(a-b+c)(-b-c+a)(a^5-3a^4b-a^4c+2a^3b^2-4a^3bc-14a^3c^2+2a^2b^3+10a^2b^2c-2a^2bc^2-10a^2c^3-3ab^4-4ab^3c+10ab^2c^2+4abc^3-7ac^4+b^5-b^4c-2b^3c^2+2b^2c^3+bc^4-c^5)xy^2+(2(a+b-c))(a-b+c)(-b-c+a)(a^7-5a^6b-a^6c+9a^5b^2-2a^5bc-7a^5c^2-5a^4b^3+17a^4b^2c+21a^4bc^2+31a^4c^3-5a^3b^4-28a^3b^3c-10a^3b^2c^2+28a^3bc^3+15a^3c^4+9a^2b^5+17a^2b^4c-18a^2b^3c^2-34a^2b^2c^3+9a^2bc^4+17a^2c^5-5ab^6-2ab^5c+17ab^4c^2+4ab^3c^3-19ab^2c^4-2abc^5+7ac^6+b^7-b^6c-3b^5c^2+3b^4c^3+3b^3c^4-3b^2c^5-bc^6+c^7)x^2y^2+16a^4c^2(c+a+b)(-b-c+a)(a+b-c)(a-b+c)^3y^2+(a^5-3a^4b+3a^4c+2a^3b^2-8a^3bc+6a^3c^2+2a^2b^3+6a^2b^2c-2a^2bc^2-6a^2c^3-3ab^4+6ab^2c^2-3ac^4+b^5-b^4c-2b^3c^2+2b^2c^3+bc^4-c^5)^2y^4+16a^4c^2(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^2+8a^2c(a^2-2ab-4ac+b^2-c^2)(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^3+(a+b-c)^2(a-b+c)^2(-b-c+a)^2(a^2-2ab-4ac+b^2-c^2)^2x^4=0\)

 小曲边的三角形的三条曲线终于被破解,如下: 2a^2(60a^6-72a^5b-75a^4b^2-10a^4bc-35a^4c^2+108a^3b^3+4a^3b^2c-76a^3bc^2-36a^3c^3+22a^2b^4-44a^2b^3c+4a^2b^2c^2+36a^2bc^3-18a^2c^4-36ab^5+36ab^4c+72ab^3c^2-72ab^2c^3-36abc^4+36ac^5-7b^6+14b^5c+7b^4c^2-28b^3c^3+7b^2c^4+14bc^5-7c^6)(a-b+c)^3(a+b-c)^5(a-b-c)^6x0^4y0^2+a^4(a+b+c)^2(a+b-c)^6(a-b+c)^6(a-b-c)^6y0^4+(64(a+b-c))(a-b+c)(5a^{12}-10a^{11}b-10a^{11}c-4a^{10}b^2+44a^{10}bc-4a^{10}c^2+18a^9b^3-34a^9b^2c-34a^9bc^2+18a^9c^3-7a^8b^4-48a^8b^3c+126a^8b^2c^2-48a^8bc^3-7a^8c^4-4a^7b^5+76a^7b^4c-72a^7b^3c^2-72a^7b^2c^3+76a^7bc^4-4a^7c^5-24a^6b^5c-64a^6b^4c^2+176a^6b^3c^3-64a^6b^2c^4-24a^6bc^5+4a^5b^7-20a^5b^6c+100a^5b^5c^2-84a^5b^4c^3-84a^5b^3c^4+100a^5b^2c^5-20a^5bc^6+4a^5c^7+7a^4b^8+16a^4b^7c-76a^4b^6c^2-16a^4b^5c^3+138a^4b^4c^4-16a^4b^3c^5-76a^4b^2c^6+16a^4bc^7+7a^4c^8-18a^3b^9-2a^3b^8c+56a^3b^7c^2+24a^3b^6c^3-60a^3b^5c^4-60a^3b^4c^5+24a^3b^3c^6+56a^3b^2c^7-2a^3bc^8-18a^3c^9+4a^2b^{10}+12a^2b^9c-12a^2b^8c^2-48a^2b^7c^3+8a^2b^6c^4+72a^2b^5c^5+8a^2b^4c^6-48a^2b^3c^7-12a^2b^2c^8+12a^2bc^9+4a^2c^{10}+10ab^{11}-10ab^{10}c-50ab^9c^2+50ab^8c^3+100ab^7c^4-100ab^6c^5-100ab^5c^6+100ab^4c^7+50ab^3c^8-50ab^2c^9-10abc^{10}+10ac^{11}-5b^{12}+30b^{10}c^2-75b^8c^4+100b^6c^6-75b^4c^8+30b^2c^{10}-5c^{12})(a-b-c)^2x0^6y0^6-(32(3a+b-c))(a+b-c)^4(a-b+c)^4(a-b-c)^8x0^{11}+(321a^4+516a^3b-516a^3c+246a^2b^2-492a^2bc+246a^2c^2+36ab^3-108ab^2c+108abc^2-36ac^3+b^4-4b^3c+6b^2c^2-4bc^3+c^4)(a+b-c)^4(a-b+c)^4(a-b-c)^8x0^8+(8(a-b+c))(4a^9-4a^8c-3a^7b^2+6a^7bc-7a^7c^2-5a^6b^3+19a^6b^2c-11a^6bc^2-3a^6c^3-9a^5b^4-16a^5b^3c+18a^5b^2c^2-16a^5bc^3+23a^5c^4+9a^4b^5-21a^4b^4c+30a^4b^3c^2-38a^4b^2c^3+25a^4bc^4-5a^4c^5+11a^3b^6+10a^3b^5c-39a^3b^4c^2-20a^3b^3c^3+45a^3b^2c^4+10a^3bc^5-17a^3c^6-3a^2b^7+5a^2b^6c-7a^2b^5c^2+a^2b^4c^3+23a^2b^3c^4-17a^2b^2c^5-13a^2bc^6+11a^2c^7-3ab^8+12ab^6c^2-18ab^4c^4+12ab^2c^6-3ac^8-b^9+b^8c+4b^7c^2-4b^6c^3-6b^5c^4+6b^4c^5+4b^3c^6-4b^2c^7-bc^8+c^9)(a^4-2a^2b^2+4a^2bc-2a^2c^2+b^4-2b^2c^2+c^4)^2y0^{10}+(16(15a^16-40a^15b-40a^15c-24a^14b^2+224a^14bc-24a^14c^2+136a^{13}b^3-232a^{13}b^2c-232a^{13}bc^2+136a^{13}c^3-28a^{12}b^4-448a^{12}b^3c+1048a^{12}b^2c^2-448a^{12}bc^3-28a^{12}c^4-168a^{11}b^5+824a^{11}b^4c-656a^{11}b^3c^2-656a^{11}b^2c^3+824a^{11}bc^4-168a^{11}c^5+88a^{10}b^6+32a^{10}b^5c-1496a^{10}b^4c^2+2752a^{10}b^3c^3-1496a^{10}b^2c^4+32a^{10}bc^5+88a^{10}c^6+72a^9b^7-648a^9b^6c+1768a^9b^5c^2-1192a^9b^4c^3-1192a^9b^3c^4+1768a^9b^2c^5-648a^9bc^6+72a^9c^7-102a^8b^8+384a^8b^7c-40a^8b^6c^2-1920a^8b^5c^3+3356a^8b^4c^4-1920a^8b^3c^5-40a^8b^2c^6+384a^8bc^7-102a^8c^8+72a^7b^9-120a^7b^8c-608a^7b^7c^2+1824a^7b^6c^3-1168a^7b^5c^4-1168a^7b^4c^5+1824a^7b^3c^6-608a^7b^2c^7-120a^7bc^8+72a^7c^9+88a^6b^{10}-224a^6b^9c+376a^6b^8c^2-384a^6b^7c^3-464a^6b^6c^4+1216a^6b^5c^5-464a^6b^4c^6-384a^6b^3c^7+376a^6b^2c^8-224a^6bc^9+88a^6c^{10}-168a^5b^{11}+328a^5b^{10}c-24a^5b^9c^2-456a^5b^8c^3+752a^5b^7c^4-432a^5b^6c^5-432a^5b^5c^6+752a^5b^4c^7-456a^5b^3c^8-24a^5b^2c^9+328a^5bc^{10}-168a^5c^{11}-28a^4b^{12}+64a^4b^{11}c+136a^4b^{10}c^2-192a^4b^9c^3-292a^4b^8c^4+128a^4b^7c^5+368a^4b^6c^6+128a^4b^5c^7-292a^4b^4c^8-192a^4b^3c^9+136a^4b^2c^{10}+64a^4bc^{11}-28a^4c^{12}+136a^3b^{13}-152a^3b^{12}c-528a^3b^{11}c^2+624a^3b^{10}c^3+600a^3b^9c^4-840a^3b^8c^5+160a^3b^7c^6+160a^3b^6c^7-840a^3b^5c^8+600a^3b^4c^9+624a^3b^3c^{10}-528a^3b^2c^{11}-152a^3bc^{12}+136a^3c^{13}-24a^2b^14-32a^2b^{13}c+120a^2b^{12}c^2+192a^2b^{11}c^3-216a^2b^{10}c^4-480a^2b^9c^5+120a^2b^8c^6+640a^2b^7c^7+120a^2b^6c^8-480a^2b^5c^9-216a^2b^4c^{10}+192a^2b^3c^{11}+120a^2b^2c^{12}-32a^2bc^{13}-24a^2c^14-40ab^15+40ab^14c+280ab^{13}c^2-280ab^{12}c^3-840ab^{11}c^4+840ab^{10}c^5+1400ab^9c^6-1400ab^8c^7-1400ab^7c^8+1400ab^6c^9+840ab^5c^{10}-840ab^4c^{11}-280ab^3c^{12}+280ab^2c^{13}+40abc^14-40ac^15+15b^16-120b^14c^2+420b^{12}c^4-840b^{10}c^6+1050b^8c^8-840b^6c^{10}+420b^4c^{12}-120b^2c^14+15c^16))x0^4y0^8+a^2(a+b+c)(9a^{10}-18a^9b+14a^9c-7a^8b^2-26a^8bc+13a^8c^2+40a^7b^3-40a^7b^2c-8a^7bc^2+8a^7c^3-25a^6b^4+76a^6b^3c-34a^6b^2c^2+52a^6bc^3-69a^6c^4-22a^5b^5+50a^5b^4c-52a^5b^3c^2-4a^5b^2c^3+74a^5bc^4-46a^5c^5+37a^4b^6-94a^4b^5c+107a^4b^4c^2-68a^4b^3c^3+59a^4b^2c^4-94a^4bc^5+53a^4c^6-4a^3b^7-20a^3b^6c+12a^3b^5c^2+60a^3b^4c^3-12a^3b^3c^4-60a^3b^2c^5+4a^3bc^6+20a^3c^7-16a^2b^8+48a^2b^7c-16a^2b^6c^2-80a^2b^5c^3+80a^2b^4c^4+16a^2b^3c^5-48a^2b^2c^6+16a^2bc^7+4ab^9-4ab^8c-16ab^7c^2+16ab^6c^3+24ab^5c^4-24ab^4c^5-16ab^3c^6+16ab^2c^7+4abc^8-4ac^9+2b^{10}-4b^9c-6b^8c^2+16b^7c^3+4b^6c^4-24b^5c^5+4b^4c^6+16b^3c^7-6b^2c^8-4bc^9+2c^{10})(a+b-c)^3(a-b+c)^3(a-b-c)^3y0^6+(32(3a^8-4a^7b-4a^7c-4a^6b^2+16a^6bc-4a^6c^2+4a^5b^3-4a^5b^2c-4a^5bc^2+4a^5c^3+2a^4b^4-16a^4b^3c+28a^4b^2c^2-16a^4bc^3+2a^4c^4+4a^3b^5+4a^3b^4c-8a^3b^3c^2-8a^3b^2c^3+4a^3bc^4+4a^3c^5-4a^2b^6+4a^2b^4c^2+4a^2b^2c^4-4a^2c^6-4ab^7+4ab^6c+12ab^5c^2-12ab^4c^3-12ab^3c^4+12ab^2c^5+4abc^6-4ac^7+3b^8-12b^6c^2+18b^4c^4-12b^2c^6+3c^8))(a^4-2a^2b^2+4a^2bc-2a^2c^2+b^4-2b^2c^2+c^4)^2x0^2y0^{10}+16(a^4-2a^2b^2+4a^2bc-2a^2c^2+b^4-2b^2c^2+c^4)^4y0^{12}+a^2(67a^9-131a^8b+35a^8c-78a^7b^2-164a^7bc-94a^7c^2+318a^6b^3+102a^6b^2c-86a^6bc^2-110a^6c^3-83a^5b^4+108a^5b^3c+334a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8c^2-24b^7c^3-6b^6c^4+36b^5c^5-6b^4c^6-24b^3c^7+9b^2c^8+6bc^9-3c^{10})(a+b-c)^3(a-b+c)^3(a-b-c)^4x0^3y0^4-16a(3a+b-c)(15a^2+12ab-12ac+b^2-2bc+c^2)(a-b+c)^4(a+b-c)^5(a-b-c)^8x0^7-4a^3(a+b+c)(5a^6-34a^5c-11a^4b^2-8a^4bc+51a^4c^2+44a^3b^2c+48a^3bc^2-28a^3c^3+7a^2b^4+8a^2b^3c-38a^2b^2c^2-8a^2bc^3+31a^2c^4-10ab^4c+20ab^2c^3-10ac^5-b^6+3b^4c^2-3b^2c^4+c^6)(a-b-c)^4(a+b-c)^4(a-b+c)^5x0y0^4=0 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 @数学星空. 这个太长了，有点恐怖，，　虽然计算出来很不容易，但缺少了美感，不好玩，我们要老胡的美妙作图～

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