X498

YFF CONCURRENT CONGRUENT CIRCLES POINT

X498 and X499 are harmonic conjugate points with respect to X1 and X2, in analogy with such pairs with respect to X1, X4 and with respect to X1, X5.
< 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, 12 }: Central Line L(3, 12) { 4, 35 }: Central Line L(4, 35) { 5, 55 }: Central Line L(5, 55) { 37, 91 }: Central Line L(37, 91) { 56, 140 }: Central Line L(56, 140) 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_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_4 = ORTHOCENTER (0.6122, 0.3958) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_91 = ISOGONAL CONJUGATE OF X(47) (-0.0865, -0.0205) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_498 = YFF CONCURRENT CONGRUENT CIRCLES POINT (0.5518, 0.2140) A B C X1 X2 X3 X12 X4 X35 X5 X55 X37 X91 X56 X140 X498 (a, b, c) = (5, 6, 7) | AB = 1

X498 (YFF CONCURRENT CONGRUENT CIRCLES POINT) 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): \(2.071652786830\)
▸ Alternate Coordinates1 form

  • Trilinears: \(bc + 2R^{2} : :\)

▸ Linear Combinations1 combination
▸ Incident Central Lines14 lines

{1, 2}, {3, 12}, {4, 35}, {5, 55}, {9, 920}, {36, 388}, {37, 91}, {46, 226}, {47, 171}, {56, 140}, {141, 611}, {191, 329}, {255, 750}, {345, 1089}

▸ Functional Relations1 relation

  • X498 = homothetic center of anti-Euler triangle and (cross-triangle of ABC and 2nd isogonal triangle of X1)

Notes

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