X1276

2nd EVANS PERSPECTOR

Evans conjectured that X1, X484, X1276, X1277 are concyclic, and he reported that Paul Yiu confirmed this conjecture and noted that the center of this circle is X1019. (Lawrence Evans, 2/24/2003)
< 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) { 4, 9 }: Orthocenter–Spieker Line { 1, 15 }: Central Line L(1, 15) { 14, 484 }: Central Line L(14, 484) { 63, 616 }: Central Line L(63, 616) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_9 = MITTENPUNKT (0.5110, 0.1615) X_1 = INCENTER (0.5714, 0.2333) X_15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) X_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_484 = 1st EVANS PERSPECTOR (2.0952, 3.0327) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_616 = ANTICOMPLEMENT OF X(13) (0.4364, 0.0360) X_1276 = 2nd EVANS PERSPECTOR (4.3665, 9.0809) A B C X4 X9 X1 X15 X14 X484 X63 X616 X1276 (a, b, c) = (5, 6, 7) | AB = 1

X1276 (2nd EVANS PERSPECTOR) 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 \(f(A,B,C) = \sqrt{3/4} + \sin(A + \pi /3) - \sin(B + \pi /3) - \sin(C + \pi /3)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(-24.757479872179\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\sqrt{3/4} + \cos(A - \pi /6) - \cos(B - \pi /6) - \cos(C - \pi /6)\)

X1276 lies on the Neuberg cubic.

▸ Incident Central Lines4 lines

{1, 15}, {4, 9}, {14, 484}, {63, 616}

▸ Functional Relations4 relations

Notes

Let T be the excentral triangle, whose vertices are the A-, B-, C- excenters of triangle ABC. Let U be the equilateral triangle having segment BC as base with vertex A' on the side of BC that does not contain vertex A. Define B' and C' cyclically, and let T' be the triangle A'B'C'. Let V be the equilateral triangle having BC as base with vertex A″ on the side of BC that contains A. Define B″ and C″ cyclically, and let T″ = A″B″C″. Then T and T' are perspective, and X1276 is their perspector. (Lawrence Evans, 2/4/2003)

X1276 is the perspector of the excentral triangle and the apices of equilateral triangles constructed outward from the sides, as in the construction of X13. More generally, the excentral triangle is perspective to every Kiepert triangle. The locus of the perspector Kθ is the line X4X9. Specifically, Kθ divides the segment from X75 to X9 in the ratio - ((4R+r)/s) cot θ. (Paul Yiu, 2/27/04).