X2904

ORTHIC-ISOGONAL CONJUGATE OF X24

Let P be an arbitrary point of the circumcircle , O, and let S(P) be the Simson-Wallace line of P, and P' the reflection of P in S(P). The locus of P' as P ranges through O is a bicircular circumquartic that meets the sidelines of ABC in 6 points other than A, B, C. The 6 points lie on a conic, here called the Montesdeoca conic, M. The center of M is X2904, and the perspector of M is X14111. Let u = \(S_{A}\), v = \(S_{B}\), w = \(S_{C}\), \(f(a,b,c) = u^{2}(S^{2} - v^{2})(S^{2} - w^{2})\), \(g(a,b,c) = 2vw(u^{2} - S^{2})(S^{2} + vw)\). A barycentric equation for M is
< 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, 70 }: Central Line L(6, 70) { 24, 52 }: Central Line L(24, 52) { 25, 195 }: Central Line L(25, 195) { 155, 403 }: Central Line L(155, 403) { 185, 378 }: Central Line L(185, 378) { 156, 1112 }: Central Line L(156, 1112) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.6627, 0.5278) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_70 = ISOGONAL CONJUGATE OF X(26) (0.7750, -1.1016) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) X_195 = -CEVA CONJUGATE OF X(3) (0.6761, 0.3331) X_155 = EIGENCENTER OF ORTHIC TRIANGLE (1.1658, 0.5033) X_403 = OF THE ORTHIC TRIANGLE (0.7040, 0.6358) X_185 = NAGEL POINT OF THE ORTHIC TRIANGLE (0.6033, 0.1643) X_378 = REFLECTION OF X(22) IN X(3) (0.5857, 0.3262) X_156 = -OF-TANGENTIAL-TRIANGLE (0.6520, -0.1707) X_1112 = CROSSPOINT OF X(4) AND X(250) (0.5737, 0.4649) X_2904 = ORTHIC-ISOGONAL CONJUGATE OF X(24) (0.5471, 0.6807) A B C X4 X24 X6 X70 X52 X25 X195 X155 X403 X185 X378 X156 X1112 X2904 (a, b, c) = (5, 6, 7) | AB = 1

X2904 (ORTHIC-ISOGONAL CONJUGATE OF X24) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where f(a,b,c) is as described just before X2883
Barycentrics: \(a^{2}(a^{4} + b^{4} + c^{4} - 2a^{2}b^{2} - 2a^{2}c^{2})[a^{6} - 3a^{4}(b^{2} + c^{2}) + 3a^{2}(b^{4} + c^{4}) - (b^{2} - c^{2})^{2}(b^{2} + c^{2})]/(b^{2} + c^{2} - a^{2}) : :\)
Search Value (6, 9, 13): \(-1.962230807342\)
▸ Incident Central Lines11 lines

{3, 1986}, {4, 1994}, {6, 70}, {24, 52}, {25, 195}, {155, 403}, {156, 1112}, {185, 378}, {576, 1843}, {648, 847}, {2888, 2914}

▸ Functional Relations2 relations

Notes

Let O be the circumcircle of a triangle ABC. Let

  • A' = perpendicular bisector of segment OA, and define B' and C' cyclically

  • A″ = B'∩C', and define B″ and C″ cyclically

  • A* = reflection of A* in BC, and define B* and C* cyclically.

Then A*B*C* is perspective to the orthic triangle of ABC, and the perspector is X2904. (Angel Montesdeoca, November 12, 2016).

f(a,b,c)\(x^{2}\) + f(b,c,a)\(y^{2}\) + (f,c,a,b)\(z^{2}\) + g(a,b,c)yz + g(b,c,a)zx + g(c,b,a)xy = 0.

See HG121017 and X2904.