X1677

EXSIMILICENTER(1st LEMOINE CIRCLE, NINE-POINT CIRCLE)

X1677 is the exsimilicenter(1st Lemoine circle, nine-point circle).
< 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) { 5, 182 }: Central Line L(5, 182) { 2, 1342 }: Central Line L(2, 1342) { 4, 1343 }: Central Line L(4, 1343) { 11, 1673 }: Central Line L(11, 1673) { 12, 1672 }: Central Line L(12, 1672) { 76, 1670 }: Central Line L(76, 1670) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_32 = 3rd POWER POINT (0.6400, 0.3332) X_1676 = INSIMILICENTER(1st LEMOINE CIRCLE, NINE-POINT CIRCLE) (0.5533, 0.2188) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_182 = MIDPOINT OF BROCARD DIAMETER (0.5500, 0.1846) X_2 = CENTROID (0.5374, 0.2000) X_1342 = INSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE) (0.5319, 0.1548) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1343 = EXSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE) (0.6155, 0.2927) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_1673 = EXSIMILICENTER(INCIRCLE, 1st LEMOINE CIRCLE) (0.6607, 0.4358) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_1672 = INSIMILICENTER(INCIRCLE, 1st LEMOINE CIRCLE) (0.5619, 0.2116) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_1670 = INVERSE-IN-BROCARD-CIRCLE OF X(1342) (0.4118, -0.0436) X_1677 = EXSIMILICENTER(1st LEMOINE CIRCLE, NINE-POINT CIRCLE) (0.6019, 0.7290) A B C X32 X1676 X5 X182 X2 X1342 X4 X1343 X11 X1673 X12 X1672 X76 X1670 X1677 (a, b, c) = (5, 6, 7) | AB = 1

X1677 (EXSIMILICENTER(1st LEMOINE CIRCLE, NINE-POINT CIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\sec(A + \omega /2) : :\)
Search Value (6, 9, 13): \(-61.957429814795\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\cos(B - C) - \cos(A - \omega ) : :\)

▸ Linear Combinations1 combination

X1677 lies on the Kiepert hyperbola.

▸ Incident Central Lines19 lines

{2, 1342}, {4, 1343}, {5, 182}, {10, 1683}, {11, 1673}, {12, 1672}, {76, 1670}, {115, 2036}, {262, 1671}, {371, 2009}, {372, 2010}, {485, 1687}, {486, 1688}, {1329, 1681}, {1506, 2035}, {1662, 2039}, {1663, 2040}, {1698, 1700}, {1699, 1701}

▸ Functional Relations2 relations

Notes

Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that ∠A'BC = ∠A'CB = ω/2. Define B' and C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X1677. (Randy Hutson, July 20, 2016)