X3054

CENTER OF EVANS CONIC

X3054 = midpoint of X590 and X615
< 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) { 590, 615 }: Centroid–Symmedian Line { 2, 6 }: Centroid–Symmedian Line { 5, 187 }: Central Line L(5, 187) { 53, 468 }: Central Line L(53, 468) { 115, 549 }: Central Line L(115, 549) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_590 = ISOGONAL CONJUGATE OF X(588) (0.5615, 0.2258) X_615 = ISOGONAL CONJUGATE OF X(589) (0.4337, 0.0885) X_2 = CENTROID (0.5374, 0.2000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (1.1986, 1.2559) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6164, 0.4390) X_468 = -LINE CONJUGATE OF X(3) (0.7498, 0.7558) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_549 = MIDPOINT OF X(2) AND X(3) (0.5187, 0.1510) X_3054 = CENTER OF EVANS CONIC (0.4976, 0.1571) A B C X590 X615 X2 X6 X5 X187 X53 X468 X115 X549 X3054 (a, b, c) = (5, 6, 7) | AB = 1

X3054 (CENTER OF EVANS CONIC) 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 \(f(a,b,c) = bc(4a^{4} + 3b^{4} + 3c^{4} - 5a^{2}b^{2} - 5a^{2}c^{2} - 6b^{2}c^{2})\)
Barycentrics: \(12\sigma - a^{2}\cot \omega : 12\sigma - b^{2}\cot \omega : 12\sigma - c^{2}\cot \omega\)
Search Value (6, 9, 13): \(-113.162937858044\)
▸ Incident Central Lines10 lines

{2, 6}, {5, 187}, {39, 632}, {53, 468}, {111, 930}, {115, 549}, {140, 574}, {216, 2493}, {626, 2031}, {1384, 1656}

▸ Functional Relations3 relations

Notes

See X13 for a discussion of the Evans conic.

X3054 and X3055 were contributed by Peter J. C. Moses, Jan 14, 2005.