X478

CENTER OF YIU CONIC

Center of the Yiu conic, which passes through the points outside the circumcircle at which the excircles of ABC are tangent to the sidelines of ABC. See Paul Yiu’s
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 6, 19 }: Central Line L(6, 19) { 222, 226 }: Central Line L(222, 226) { 109, 573 }: Central Line L(109, 573) { 69, 651 }: Central Line L(69, 651) { 198, 577 }: Central Line L(198, 577) { 9, 1038 }: Central Line L(9, 1038) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_19 = CLAWSON POINT (0.6246, 0.4244) X_222 = -CEVA CONJUGATE OF X(56) (0.5653, 0.1568) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.3636, 0.5938) X_573 = ORTHOHARMONIC OF X(58) (0.4583, 0.2009) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_651 = TRILINEAR POLE OF LINE X(1)X(3) (0.3048, 0.5599) X_198 = -CEVA CONJUGATE OF X(6) (0.4741, -0.0381) X_577 = HARMONIC OF X(216) (0.4663, 0.0464) X_9 = MITTENPUNKT (0.5110, 0.1615) X_1038 = INTERSECTION OF LINES X(1)X(3) AND X(2)X(34) (0.5104, 0.1211) X_478 = CENTER OF YIU CONIC (0.5030, -0.3520) A B C X6 X19 X222 X226 X109 X573 X69 X651 X198 X577 X9 X1038 X478 (a, b, c) = (5, 6, 7) | AB = 1

X478 (CENTER OF YIU CONIC) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a[a^{4} - 2abc(b + c - a) - (b^{2} - c^{2})^{2}]/(b + c - a) : :\)
Barycentrics: \(a^{2} (a + b - c) \cdot (a - b + c) \cdot (a^{4} - b^{4} + 2 a^{2} b c - 2 a b^{2} c - 2 a b c^{2} + 2 b^{2} c^{2} - c^{4})\)
Search Value (6, 9, 13): \(-0.295568314180\)
▸ Incident Central Lines34 lines

{2, 8048}, {6, 19}, {9, 1038}, {12, 20029}, {37, 1455}, {56, 5019}, {69, 651}, {73, 2268}, {109, 573}, {154, 3195}, {198, 577}, {219, 4559}, {222, 226}, {223, 10319}, {388, 2298}, {572, 10571}, {603, 1400}, {604, 1457}, {1035, 18591}, {1122, 1407}, {1211, 5783}, {1413, 1903}, {1470, 2277}, {1766, 21147}, {2092, 11509}, {2122, 5750}, {2183, 2199}, {3142, 3330}, {3157, 10441}, {3596, 6648}, {3713, 9370}, {8231, 13388}, {8750, 18621}, {11496, 14749}

▸ Functional Relations9 relations

Notes

Let Ea be the ellipse through A having foci B and C, and define Eb and Ec cyclically. The minor vertices, A1, A2; B1, B2; C1, C2 are the touchpoints of the excircle with sidelines external to the circumcircle; i.e., they lie on the Yiu conic, centered at X478. See Figure. If ABC is a right triangle, then the Yiu conic is a pair of lines meeting in X478. Figure. (Dan Reznik, December 13, 2021)