X499

{X1,X2}-HARMONIC CONJUGATE OF X498

X499 is the harmonic conjugate of X498 with respect to X1 and X2.
< 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, 2 }: Nagel Line { 3, 11 }: Central Line L(3, 11) { 4, 36 }: Central Line L(4, 36) { 5, 56 }: Central Line L(5, 56) { 55, 140 }: Central Line L(55, 140) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_498 = YFF CONCURRENT CONGRUENT CIRCLES POINT (0.5518, 0.2140) X_1 = INCENTER (0.5714, 0.2333) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_4 = ORTHOCENTER (0.6122, 0.3958) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_499 = {X(1),X(2)}-HARMONIC CONJUGATE OF X(498) (0.4458, 0.1102) A B C X498 X1 X2 X3 X11 X4 X36 X5 X56 X55 X140 X499 (a, b, c) = (5, 6, 7) | AB = 1

X499 ({X1,X2}-HARMONIC CONJUGATE OF X498) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1 - 2 \sin B \sin C : 1 - 2 \sin C \sin A : 1 - 2 \sin A \sin B\)
Barycentrics: \((\sin A)(1 - 2 \sin B \sin C) : (\sin B)(1 - 2 \sin C \sin A) : (\sin C)(1 - 2 \sin A \sin B)\)
Search Value (6, 9, 13): \(-0.340092102503\)
▸ Alternate Coordinates1 form

  • Trilinears: \(bc - 2R^{2} : :\)

▸ Linear Combinations1 combination
▸ Incident Central Lines17 lines

{1, 2}, {3, 11}, {4, 36}, {5, 56}, {12, 999}, {17, 202}, {18, 203}, {35, 497}, {46, 946}, {47, 238}, {55, 140}, {57, 920}, {80, 944}, {141, 613}, {255, 748}, {348, 1111}, {484, 962}

Notes

Peter Yff, “Three concurrent congruent circles ‘inscribed’ in a triangle,” manuscript, 1998; X498 is the point S' on page 6.