X962

LONGUET-HIGGINS POINT

X962 is the radical center of the circles centered at A, B, C, with respective
< 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, 7 }: Soddy Line { 4, 8 }: Fuhrmann Line { 2, 40 }: Central Line L(2, 40) { 149, 151 }: Central Line L(149, 151) { 55, 411 }: Central Line L(55, 411) { 65, 497 }: Central Line L(65, 497) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_963 = ISOGONAL CONJUGATE OF X(962) (1.3582, 0.0792) X_1 = INCENTER (0.5714, 0.2333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_4 = ORTHOCENTER (0.6122, 0.3958) X_8 = NAGEL POINT (0.4694, 0.1333) X_2 = CENTROID (0.5374, 0.2000) X_40 = BEVAN POINT (0.4286, -0.0292) X_149 = REFLECTION OF X(20) IN X(104) (1.3265, -0.3333) X_151 = REFLECTION OF X(20) IN X(109) (0.3395, 1.3793) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_411 = EULER X(21)-2nd-SUBSTITUTION POINT (0.4414, -0.0514) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_497 = CROSSPOINT OF GERGONNE POINT AND NAGEL POINT (0.5593, 0.1851) X_962 = LONGUET-HIGGINS POINT (0.7551, 0.6582) A B C X963 X1 X7 X4 X8 X2 X40 X149 X151 X55 X411 X65 X497 X962 (a, b, c) = (5, 6, 7) | AB = 1

X962 (LONGUET-HIGGINS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc[a^{4} + 2a^{3}(b + c) - 4a^{2}bc - (b + c)(b - c)^{2}(2a + b + c)] : :\)
Search Value (6, 9, 13): \(-15.860728180405\)
▸ Alternate Coordinates4 forms

  • Trilinears: \(\sin B \sin C + \cos B + \cos C - \cos A - 1 : :\)

  • Trilinears: \(\cos B + \cos C + \cos B \cos C - 1 : :\)

  • Trilinears: \(1 - 2 \cos^{2}(B/2) \cos^{2}(C/2) : :\)

  • Trilinears: \(\cos A' : :\), where A' is the angle formed by the tangents from X1 to the A-excircle

▸ Incident Central Lines14 lines

{1, 7}, {2, 40}, {4, 8}, {30, 944}, {55, 411}, {65, 497}, {145, 515}, {149, 151}, {165, 1125}, {278, 412}, {382, 952}, {392, 443}, {484, 499}, {942, 1058}

▸ Functional Relations10 relations

Notes

X962 is shown in Michael S. Longuet-Higgins, “On the principal centers of a triangle,” Elemente der Mathematik 56 (2001) 122-129, to complete a simple pattern of collinearities.

Let I = X1 and H = X4 in a triangle ABC. Let Lb be the perpendicular to B at I, and let Ic be the perpendicular to CI at I. Let Ab = Lb∩AH and Ac = Lc∩AH. Let U be the circle {I, Ab, Ac}}. Let Mb be the point, other than I, where BI meets U, and let Mc be the point, other than I, where CI meets U. Let ℓa be the line MbMc, and define ℓb and ℓc cyclically. The lines ℓa, ℓb, ℓc concur in X962. (Angel Montesdeoca, March 12, 2020)

radii |CA| + |AB|, |AB| + |BC|, |BC| + |CA|. See

Floor van Lamoen, Problem 10734, American Mathematical Monthly 107 (2000) 658-659.