Center of conic has first barycentric : (a^28 - 11 a^26 (b^2 + c^2) + a^24 (37 b^4 + 84 b^2 c^2 + 37 c^4) - 2 a^22 (4 b^6 + 67 b^4 c^2 + 67 b^2 c^4 + 4 c^6) - (b^2 - c^2)^4 (b^2 + c^2)^6 (b^8 - 4 b^6 c^2 + 5 b^4 c^4 - 4 b^2 c^6 + c^8) + a^4 (b^4 - c^4)^4 (17 b^8 - 73 b^6 c^2 + 25 b^4 c^4 - 73 b^2 c^6 + 17 c^8) - 3 a^20 (51 b^8 + 71 b^6 c^2 + 29 b^4 c^4 + 71 b^2 c^6 + 51 c^8) + 3 a^18 (45 b^10 + 143 b^8 c^2 + 75 b^6 c^4 + 75 b^4 c^6 + 143 b^2 c^8 + 45 c^10) + a^2 (b^2 - c^2)^2 (b^2 + c^2)^5 (b^12 + 12 b^10 c^2 - 62 b^8 c^4 + 94 b^6 c^6 - 62 b^4 c^8 + 12 b^2 c^10 + c^12) - 2 a^6 (b^2 - c^2)^2 (b^2 + c^2)^3 (24 b^12 - 41 b^10 c^2 + 33 b^8 c^4 - 132 b^6 c^6 + 33 b^4 c^8 - 41 b^2 c^10 + 24 c^12) + a^16 (213 b^12 + 274 b^10 c^2 + 604 b^8 c^4 + 1126 b^6 c^6 + 604 b^4 c^8 + 274 b^2 c^10 + 213 c^12) + a^14 (-248 b^14 - 284 b^12 c^2 + 318 b^10 c^4 + 78 b^8 c^6 + 78 b^6 c^8 + 318 b^4 c^10 - 284 b^2 c^12 - 248 c^14) - a^8 (b^2 + c^2)^2 (9 b^16 - 194 b^14 c^2 + 74 b^12 c^4 + 458 b^10 c^6 - 502 b^8 c^8 + 458 b^6 c^10 + 74 b^4 c^12 - 194 b^2 c^14 + 9 c^16) - a^12 (105 b^16 + 250 b^14 c^2 + 822 b^12 c^4 + 710 b^10 c^6 + 50 b^8 c^8 + 710 b^6 c^10 + 822 b^4 c^12 + 250 b^2 c^14 + 105 c^16) + a^10 (179 b^18 - 49 b^16 c^2 - 496 b^14 c^4 - 340 b^12 c^6 - 766 b^10 c^8 - 766 b^8 c^10 - 340 b^6 c^12 - 496 b^4 c^14 - 49 b^2 c^16 + 179 c^18):...:...)
Let H=X(4)-Orthocenter of ABC. A 1 B 1 C 1 cevian triangle of H. B1 A , C1 A are reflections of B1, C1 on A. L A , line through B1 A , C1...
Center of conic has first barycentric : (a^28 - 11 a^26 (b^2 + c^2) + a^24 (37 b^4 + 84 b^2 c^2 + 37 c^4) - 2 a^22 (4 b^6 + 67 b^4 c^2 + 67 b^2 c^4 + 4 c^6) - (b^2 - c^2)^4 (b^2 + c^2)^6 (b^8 - 4 b^6 c^2 + 5 b^4 c^4 - 4 b^2 c^6 +
ReplyDeletec^8) + a^4 (b^4 - c^4)^4 (17 b^8 - 73 b^6 c^2 + 25 b^4 c^4 -
73 b^2 c^6 + 17 c^8) -
3 a^20 (51 b^8 + 71 b^6 c^2 + 29 b^4 c^4 + 71 b^2 c^6 + 51 c^8) +
3 a^18 (45 b^10 + 143 b^8 c^2 + 75 b^6 c^4 + 75 b^4 c^6 + 143 b^2 c^8 + 45 c^10) + a^2 (b^2 - c^2)^2 (b^2 + c^2)^5 (b^12 + 12 b^10 c^2 - 62 b^8 c^4 + 94 b^6 c^6 - 62 b^4 c^8 + 12 b^2 c^10 + c^12) - 2 a^6 (b^2 - c^2)^2 (b^2 + c^2)^3 (24 b^12 - 41 b^10 c^2 + 33 b^8 c^4 - 132 b^6 c^6 + 33 b^4 c^8 - 41 b^2 c^10 + 24 c^12) + a^16 (213 b^12 + 274 b^10 c^2 + 604 b^8 c^4 + 1126 b^6 c^6 + 604 b^4 c^8 + 274 b^2 c^10 + 213 c^12) + a^14 (-248 b^14 - 284 b^12 c^2 + 318 b^10 c^4 + 78 b^8 c^6 + 78 b^6 c^8 + 318 b^4 c^10 - 284 b^2 c^12 - 248 c^14) - a^8 (b^2 + c^2)^2 (9 b^16 - 194 b^14 c^2 + 74 b^12 c^4 + 458 b^10 c^6 - 502 b^8 c^8 + 458 b^6 c^10 + 74 b^4 c^12 - 194 b^2 c^14 + 9 c^16) - a^12 (105 b^16 + 250 b^14 c^2 + 822 b^12 c^4 + 710 b^10 c^6 + 50 b^8 c^8 + 710 b^6 c^10 + 822 b^4 c^12 + 250 b^2 c^14 + 105 c^16) + a^10 (179 b^18 - 49 b^16 c^2 - 496 b^14 c^4 - 340 b^12 c^6 - 766 b^10 c^8 - 766 b^8 c^10 - 340 b^6 c^12 - 496 b^4 c^14 - 49 b^2 c^16 + 179 c^18):...:...)