X1659

YIU-PAASCHE POINT

X1659 = X482-cross conjugate of X7
< 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, 4 }: Incenter–Orthocenter Line { 2, 176 }: Central Line L(2, 176) { 57, 482 }: Central Line L(57, 482) { 75, 491 }: Central Line L(75, 491) { 92, 1585 }: Central Line L(92, 1585) { 553, 1373 }: Central Line L(553, 1373) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_482 = 2nd EPPSTEIN POINT (0.5835, 0.2611) X_7 = GERGONNE POINT (0.5903, 0.2769) X_1 = INCENTER (0.5714, 0.2333) X_4 = ORTHOCENTER (0.6122, 0.3958) X_2 = CENTROID (0.5374, 0.2000) X_176 = EQUAL DETOUR POINT (0.5803, 0.2537) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_491 = CEVAPOINT OF X(2) AND X(487) (0.7449, 0.4229) X_92 = CEVAPOINT OF INCENTER AND CLAWSON POINT (0.5949, 0.3639) X_1585 = POINT CAPH I (0.5907, 0.3394) X_553 = POINT MAIA II (0.5963, 0.2857) X_1373 = 1st GRIFFITHS POINT (0.5861, 0.2673) X_1659 = YIU-PAASCHE POINT (0.5749, 0.2470) A B C X482 X7 X1 X4 X2 X176 X57 X75 X491 X92 X1585 X553 X1373 X1659 (a, b, c) = (5, 6, 7) | AB = 1

X1659 (YIU-PAASCHE POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = 1/(1 + \sin A + \cos A)\)
Barycentrics: \(\sin A + \cos B + \cos C : :\)
Search Value (6, 9, 13): \(1.303745560539\)
▸ Incident Central Lines6 lines

{1, 4}, {2, 176}, {57, 482}, {75, 491}, {92, 1585}, {553, 1373}

▸ Functional Relations3 relations

Notes

For a discussion of this point, see Paul Yiu, Introduction to the Geometry of the Triangle, 2002, Article 3.5.4 Exercise 4d.

(The preceding Exercise 4c presents the Paasche point, X1123,) (Contributed by Darij Grinberg, 8/24/03)

See Francisco Javier García Capitán, Hyacinthos #21541, 2/14/2013.