X496

{X1,X5}-HARMONIC CONJUGATE OF X495

X496 is the {X1,X5}-harmonic conjugate of X495.
< 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, 5 }: Feuerbach Line { 30, 56 }: Central Line L(30, 56) { 55, 140 }: Central Line L(55, 140) { 149, 404 }: Central Line L(149, 404) { 202, 397 }: Central Line L(202, 397) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_495 = JOHNSON MIDPOINT (0.5666, 0.2382) X_1 = INCENTER (0.5714, 0.2333) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_30 = EULER INFINITY POINT (0.7469, 0.7481) 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_149 = REFLECTION OF X(20) IN X(104) (1.3265, -0.3333) X_404 = {X(2),X(3)}-HARMONIC CONJUGATE OF X(21) (0.5538, 0.2429) X_202 = -CEVA CONJUGATE OF X(15) (0.5962, 0.2697) X_397 = CROSSPOINT OF X(4) AND X(17) (0.6029, 0.2975) X_496 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(495) (0.5843, 0.2201) A B C X495 X1 X5 X30 X56 X55 X140 X149 X404 X202 X397 X496 (a, b, c) = (5, 6, 7) | AB = 1

X496 ({X1,X5}-HARMONIC CONJUGATE OF X495) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(2 - \cos(B - C) : 2 - \cos(C - A) : 2 - \cos(A - B)\)
Barycentrics: \((\sin A)[2 - \cos(B - C)] : (\sin B)[2 - \cos(C - A)] : (\sin C)[2 - \cos(A - B)]\)
Search Value (6, 9, 13): \(2.026083201065\)
▸ Linear Combinations1 combination
▸ Incident Central Lines16 lines

{1, 5}, {2, 1058}, {3, 497}, {4, 999}, {30, 56}, {35, 549}, {36, 550}, {55, 140}, {149, 404}, {202, 397}, {203, 398}, {381, 388}, {390, 631}, {613, 1069}, {614, 1062}, {942, 946}

▸ Functional Relations3 relations

Notes

Peter Yff, “Three concurrent congruent circles ‘inscribed’ in a triangle,” manuscript, 1998; X496 is the point R' on page 5.