X3145

X226-CEVA CONJUGATE OF X6

X3145 = X226-Ceva conjugate of X6
< 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) { 2, 3 }: Euler Line { 45, 198 }: Central Line L(45, 198) { 35, 228 }: Central Line L(35, 228) { 51, 580 }: Central Line L(51, 580) { 184, 581 }: Central Line L(184, 581) { 283, 511 }: Central Line L(283, 511) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_45 = -BETH CONJUGATE OF X(1) (0.5418, 0.1981) X_198 = -CEVA CONJUGATE OF X(6) (0.4741, -0.0381) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_228 = -CEVA CONJUGATE OF X(71) (0.5231, 0.1126) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_580 = ORTHOGONAL OF X(284) (0.7089, 0.4470) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_581 = ORTHOHARMONIC OF X(284) (0.5657, 0.2106) X_283 = -CROSS CONJUGATE OF X(21) (0.4475, 0.0704) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_3145 = -CEVA CONJUGATE OF X(6) (0.5570, 0.2513) A B C X226 X6 X2 X3 X45 X198 X35 X228 X51 X580 X184 X581 X283 X511 X3145 (a, b, c) = (5, 6, 7) | AB = 1

X3145 (X226-CEVA CONJUGATE OF X6) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X226 and \(u : v : w\) = X6
Barycentrics: \(ag(a,b,c) : bg(b,c,a) : cg(c,a,b)\)
Search Value (6, 9, 13): \(1.139193227048\)
▸ Alternate Coordinates1 form

  • Trilinears: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a[(a^{5} - a^{3}(b^{2} + bc + c^{2}) + a^{2}(b^{3} + c^{3}) + abc(b + c)^{2} - (b + c)(b^{2} + bc + c^{2})(b - c)^{2}] (E\). Danneels)

▸ Incident Central Lines16 lines

{1, 1283}, {2, 3}, {35, 228}, {41, 2276}, {45, 198}, {51, 580}, {55, 976}, {56, 1626}, {109, 1425}, {184, 581}, {283, 511}, {500, 1437}, {692, 2594}, {1284, 1486}, {1400, 1950}, {1495, 2360}

▸ Functional Relations5 relations

Notes

Let P be a point in the plane of ABC and not on the circumcircle. Let (OA) be the circle tangent to the circumcircle at A and passing through P. Let A' be the antipode of A in (OA). Let LA be the tangent to (OA) at A'. Define LB and LC cyclically. Let TA = LB∩LC, and define TB, TC cyclically. Triangle TATBTC is homothetic to the tangential triangle. When P = X1, the center of homothety is X3145. (Randy Hutson, June 7, 2019)