X363

EQUAL PERIMETERS ISOSCELIZER POINT

If X = X363, the isoscelizer triangles of X have equal perimeters.
< 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) { 40, 164 }: Central Line L(40, 164) { 1, 289 }: Central Line L(1, 289) { 165, 166 }: Central Line L(165, 166) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_40 = BEVAN POINT (0.4286, -0.0292) X_164 = INCENTER OF EXCENTRAL TRIANGLE (0.4930, 0.1115) X_1 = INCENTER (0.5714, 0.2333) X_289 = ISOGONAL CONJUGATE OF X(236) (0.5893, 0.2483) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_166 = GERGONNE POINT OF EXCENTRAL TRIANGLE (0.5156, 0.1554) X_363 = EQUAL PERIMETERS ISOSCELIZER POINT (0.5347, 0.2024) A B C X40 X164 X1 X289 X165 X166 X363 (a, b, c) = (5, 6, 7) | AB = 1

X363 (EQUAL PERIMETERS ISOSCELIZER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(b/(1 + \sin B/2) + c/(1 + \sin C/2) - a/(1 + \sin A/2) : :\)
Search Value (6, 9, 13): \(2.584980945018\)
▸ Incident Central Lines3 lines

{1, 289}, {40, 164}, {165, 166}

▸ Functional Relations2 relations

Notes

X363 is the homothetic center of the excentral triangle and the inner Hutson triangle. A construction of the latter follows. The internal bisector of angle A meets the A-excircle in two points. Let PA be the point closer to line BC and let QA be the other point. Define PB and PC cyclically, and define QB and QC cyclically. Let LA be the line tangent to the A-excircle at PA, and define LB and LC cyclically. Let MA be the line tangent to the A-excircle at QA, and define MB and MC cyclically. The inner Hutson triangle is the triangle A'B'C' given by A' = LB∩LC, B' = LC∩LA, C' = LA∩LB; the outer Hutson triangle is given by A″ = MB∩MC, B″ = MC∩MA, C' = MA∩MB. (Based on a description of A'B'C' by Randy Hutson, September 23, 2011)

Peter Moses (November 10, 2011) found trilinears for A'B'C' and A″B″C″. As these are central triangles, trilinears for A' and A″ suffice:

A' = \(aU_{A}(a^{2} + b^{2} + c^{2} - 2bc - 2ca - 2ab)\) + \(bU_{B}(b + c - a)\)(c + a - b) + \(cU_{C}(b + c - a)\)(a + b - c)

: \(aU_{A}(a^{2} + b^{2} - 3c^{2} + 2bc + 2ca - 2ab)\) + UB(c - a)(-a + b + c)(a - b + c) - \(cU_{C}(-a + b + c)\)(a + b - c)

: \(aU_{A}(a^{2} - 3b^{2} + c^{2} + 2bc - 2ca + 2ab)\) - \(bU_{B}(-a + b + c)\)(a - b + c) + UC(b - a)(-a + b + c)(a + b - c)

A″ = \(aU_{A}(a^{2} + b^{2} + c^{2} - 2bc - 2ca - 2ab)\) - \(bU_{B}(b + c - a)\)(c + a - b) - \(cU_{C}(b + c - a)\)(a + b - c)

: \(aU_{A}(a^{2} + b^{2} - 3c^{2} + 2bc + 2ca - 2ab)\) - UB(c - a)(-a + b + c)(a - b + c) + \(cU_{C}(-a + b + c)\)(a + b - c)

: \(aU_{A}(a^{2} - 3b^{2} + c^{2} + 2bc - 2ca + 2ab)\) + \(bU_{B}(-a + b + c)\)(a - b + c) - UC(b - a)(-a + b + c)(a + b - c),

where UA = sqrt[bc/((a - b + c)(a + b - c)] = (1/2)csc(A/2), and UB and UC are defined cyclically.

A'B'C' is perspective to the following triangles: intouch, hexyl,Yff, and the 1st and 2nd circumperp triangles. A″B″C″ is perspective to these: ABC, intouch, hexyl, Yff, and the 1st and 2nd circumperp triangles. A'B'C' is homothetic to A″B″C″, which is perspective to the excentral triangle at X168. (Peter Moses, 11/10/11)

Let UVW and U'V'W' be the tangential-midarc triangles introduced in the preamble to X8075. The inner and outer Hutson triangles introduced at X363 are the excircles-version of UVW and U'V'W'. (Randy Hutson and César Lozada, August 29, 2015)