X1697

INSIMILICENTER(BEVAN CIRCLE, INCIRCLE)

The exsimilicenter of the Bevan circle and incircle is X57.
< 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, 3 }: OI Line (Incenter–Circumcenter Line) { 8, 9 }: Central Line L(8, 9) { 63, 145 }: Central Line L(63, 145) { 10, 497 }: Central Line L(10, 497) { 219, 380 }: Central Line L(219, 380) { 212, 595 }: Central Line L(212, 595) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_8 = NAGEL POINT (0.4694, 0.1333) X_9 = MITTENPUNKT (0.5110, 0.1615) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_497 = CROSSPOINT OF GERGONNE POINT AND NAGEL POINT (0.5593, 0.1851) X_219 = -CEVA CONJUGATE OF X(55) (0.4277, 0.0625) X_380 = INTERSECTION OF LINES X(1)X(19) AND X(9)X(55) (0.5631, 0.2035) X_212 = -CEVA CONJUGATE OF X(41) (0.4719, 0.0786) X_595 = 2nd HATZIPOLAKIS-YIU POINT (0.6456, 0.3307) X_1697 = INSIMILICENTER(BEVAN CIRCLE, INCIRCLE) (0.5449, 0.1845) A B C X1 X3 X8 X9 X63 X145 X10 X497 X219 X380 X212 X595 X1697 (a, b, c) = (5, 6, 7) | AB = 1

X1697 (INSIMILICENTER(BEVAN CIRCLE, INCIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) = 3 + \cos A - \cos B - \cos C : :\)
Search Value (6, 9, 13): \(2.732161068994\)
▸ Alternate Coordinates1 form

  • Trilinears: \(-1 + \sin A/2 \cos B/2 \cos C/2 : :\)

▸ Incident Central Lines34 lines

{1, 3}, {2, 1706}, {8, 9}, {10, 497}, {11, 1698}, {12, 1699}, {33, 1831}, {43, 1682}, {63, 145}, {71, 1732}, {84, 944}, {109, 1496}, {181, 1695}, {200, 960}, {212, 595}, {219, 380}, {221, 1419}, {226, 962}, {388, 516}, {392, 936}, {519, 1776}, {580, 1497}, {1015, 1571}, {1058, 1210}, {1124, 1703}, {1317, 1768}, {1335, 1702}, {1500, 1572}, {1672, 1701}, {1673, 1700}, {1674, 1705}, {1675, 1704}, {2007, 2018}, {2008, 2017}

▸ Functional Relations11 relations

Notes

Randy Hutson (January 29, 2015) gives 3 constructions:

    1. Let TaTbTc be the mixtilinear incentral triangle. Let A″ be the cevapoint of Tb and Tc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X1697.

    1. Let TaTbTc be the mixtilinear incentral triangle. Let A″ be the trilinear product Tb*Tc, and define B″ and C″ cyclically. The lines TaA″, TbB″, TcC″ concur in X1697.

  • Let TaTbTc be the intouch triangle, A″B″C″ the extouch triangle, and A*B*C* the excentral triangle. Let OA be the circle through TaA″A*, and define OB and OCc cyclically. X1697 is the radical center of circles OA, OB, OC.