X1676

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

X1676 is the insimilicenter(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, 1343 }: Central Line L(2, 1343) { 4, 1342 }: Central Line L(4, 1342) { 11, 1672 }: Central Line L(11, 1672) { 12, 1673 }: Central Line L(12, 1673) { 76, 1671 }: Central Line L(76, 1671) 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_1677 = EXSIMILICENTER(1st LEMOINE CIRCLE, NINE-POINT CIRCLE) (0.6019, 0.7290) 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_1343 = EXSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE) (0.6155, 0.2927) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1342 = INSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE) (0.5319, 0.1548) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_1672 = INSIMILICENTER(INCIRCLE, 1st LEMOINE CIRCLE) (0.5619, 0.2116) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_1673 = EXSIMILICENTER(INCIRCLE, 1st LEMOINE CIRCLE) (0.6607, 0.4358) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_1671 = INVERSE-IN-BROCARD-CIRCLE OF X(1343) (0.5882, 0.2477) X_1676 = INSIMILICENTER(1st LEMOINE CIRCLE, NINE-POINT CIRCLE) (0.5533, 0.2188) A B C X32 X1677 X5 X182 X2 X1343 X4 X1342 X11 X1672 X12 X1673 X76 X1671 X1676 (a, b, c) = (5, 6, 7) | AB = 1

X1676 (INSIMILICENTER(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: \(\csc(A + \omega /2) : \csc(B + \omega /2) : \csc(C + \omega /2)\)
Search Value (6, 9, 13): \(2.176933945550\)
▸ Alternate Coordinates1 form

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

▸ Linear Combinations2 combinations

X1676 lies on the Kiepert hyperbola.

▸ Incident Central Lines19 lines

{2, 1343}, {4, 1342}, {5, 182}, {10, 1684}, {11, 1672}, {12, 1673}, {76, 1671}, {115, 2035}, {262, 1670}, {371, 2010}, {372, 2009}, {485, 1688}, {486, 1687}, {1329, 1680}, {1506, 2036}, {1662, 2040}, {1663, 2039}, {1698, 1701}, {1699, 1700}

▸ Functional Relations2 relations

Notes

Let Lab and Lac be the lines obtained by rotating line BC through B and C resp., by an angle of ω/2 away from A. Let A' be Lab∩Lac. Define B' and C’cyclically. The lines AA', BB', CC' concur in X1676. (Randy Hutson, September 14, 2016)