X₃₆₁₀

1st AYME-MOSES PERSPECTOR

Moses found that the locus of X such that the cevian triangle of X is perspective to the Ayme triangle is a cubic which passes through the points X_i for i = 1, 2, 19, 75, 279, 304, 346, 2184. A barycentric equation for this Ayme-Moses cubic follows:

(Cyclic sum of ayz[\(by(a^{2}\) + \(b^{2}\) - \(c^{2})\) - \(cz(a^{2}\) - \(b^{2}\) + \(c^{2}\)] ) = 0.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₃₆₁₀ (1st AYME-MOSES PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Cevian Triangle △P_aP_bP_c — vertices P_a, P_b, P_c (Traces of cevians through target center on BC, CA, AB)

• Concurrence Lines: A–P_a, B–P_b, C–P_c concurring at X₃₆₁₀

Trilinears: \(bc(b + c)(a^{2} - b^{2} - c^{2})(a^{2} + b^{2} + c^{2} + 2bc) : :\)

Search Value (6, 9, 13): \(6.068567361447\)

▸ Incident Central Lines3 lines

{10, 37}, {19, 346}, {612, 2345}

▸ Functional Relations1 relation

• X₃₆₁₀ = perspector of ABC and cross-triangle of ABC and Ayme triangle

Notes

In a Hyacinthos message dated January 10, 2011, Jean-Louis Ayme introduced a triangle as follows. Let \(R_{A}\) be the radical axis of the circumcircle and the A-excircle, and define \(R_{B}\) and \(R_{C}\) cyclically. Let T_A = \(R_{B}\)∩\(R_{C}\), and define T_B and T_C cyclically. (T_A is also the radical center of the circumcircle and the B- and C- excircles.) The Ayme triangle T_AT_BT_C is perspective to triangle ABC and also perspective to many other triangles. Peter Moses found that its perspector with the cevian triangle of X₃₄₆ is X₃₆₁₀. He also found that the A-vertex of the Ayme triangle has first barycentric as follows:

• (b + c)(\(a^{2}\) + \(b^{2}\) + \(c^{2}\) + 2bc) : \(b(a^{2}\) + \(b^{2}\) - \(c^{2})\) : \(c(a^{2}\) - \(b^{2}\) + \(c^{2})\),

from which the other two vertices are easily obtained. The Ayme triangle is perspective to ABC with perspector X₁₉.

The Ayme triangle is homothetic to the incentral triangle, and the center of homothety is X₆₁₂. (Randy Hutson, September 14, 2016)