X₃₂₇₂

CENTER OF EQUILATERAL TRIANGLE J1J2J3

Let A₁B₁C₁, A₂B₂C₂, A₃B₃C₃ be the 1st Morley, 2nd Morley and 3rd Morley triangles of ABC, respectively. Then {A, A₁, A₂, A₃} are concyclic on a circle, A0.

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

X₃₂₇₂ (CENTER OF EQUILATERAL TRIANGLE J1J2J3) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {396, 523}, {357, 1135}

• Related Centers: X₃₉₆ (MIDPOINT OF X₁₃ AND X₁₅), X₅₂₃ (ISOGONAL CONJUGATE OF X₁₁₀), X₃₅₇ (1st MORLEY-TAYLOR-MARR CENTER), X₁₁₃₅ (4th MORLEY-TAYLOR-MARR CENTER)

Trilinears: \(\cos(B/3 - C/3) : :\)

Search Value (6, 9, 13): \(1.622283534100\)

▸ Alternate Coordinates1 form

• Trilinears: \(\cos(B/3) \cos(C/3) + \sin(B/3) \sin(C/3) : :\)

▸ Incident Central Lines8 lines

{3, 3334}, {356, 3273}, {357, 1135}, {358, 1136}, {396, 523}, {1134, 1137}, {3274, 3276}, {3275, 3277}

▸ Functional Relations1 relation

• X₃₂₇₂ = {X₃₅₇,X₃₆₀₃}-harmonic conjugate of X₁₁₃₅

Notes

{B, B₁, B₂, B₃} are concyclic on a circle, B0.

{C, C₁, C₂, C₃} are concyclic on a circle C0.

The radical center of A0, B0, C0 is X₃₂₇₂. (César Lozada, June 16, 2018)