X1458

X518-BETH CONJUGATE OF X518

X1458 = crosssum of X1 and X516
< 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) { 1, 7 }: Soddy Line { 6, 41 }: Central Line L(6, 41) { 36, 59 }: Central Line L(36, 59) { 42, 57 }: Central Line L(42, 57) { 31, 222 }: Central Line L(31, 222) { 241, 518 }: Central Line L(241, 518) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_518 = ISOGONAL CONJUGATE OF X(105) (0.9154, 0.6489) X_1 = INCENTER (0.5714, 0.2333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_41 = -CEVA CONJUGATE OF X(31) (0.5823, 0.2244) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_59 = ISOGONAL CONJUGATE OF X(11) (0.5333, 0.4355) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_222 = -CEVA CONJUGATE OF X(56) (0.5653, 0.1568) X_241 = -LINE CONJUGATE OF X(55) (1.1217, 1.2442) X_1458 = -BETH CONJUGATE OF X(518) (0.8889, 0.9677) A B C X518 X1 X7 X6 X41 X36 X59 X42 X57 X31 X222 X241 X1458 (a, b, c) = (5, 6, 7) | AB = 1

X1458 (X518-BETH CONJUGATE OF X518) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((1 - \cos A)u(a,b,c) : :\), where \(u : v : w\) = X518
Barycentrics: \(a^{2} (b^{2} + c^{2} - a b - a c)/(a - b - c) : :\)
Search Value (6, 9, 13): \(-0.346158285519\)
▸ Incident Central Lines28 lines

{1, 7}, {3, 1037}, {6, 41}, {31, 222}, {36, 59}, {38, 1214}, {42, 57}, {55, 1407}, {64, 963}, {65, 1418}, {108, 1430}, {109, 840}, {185, 1208}, {223, 614}, {238, 651}, {241, 518}, {244, 1465}, {256, 1476}, {354, 1427}, {513, 663}, {672, 1362}, {919, 1416}, {942, 1254}, {976, 1038}, {999, 1064}, {1201, 1419}, {1401, 1402}, {1429, 1462}

▸ Functional Relations8 relations

Notes

Let A'B'C' and A″B″C″ be the circumcevian triangles of X3513 and X3514, resp. Let A* be the trilinear product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X1458. (Randy Hutson, June 7, 2019)