X₃₆₀₄

ISOGONAL CONJUGATE OF X₃₂₇₃

X₃₆₀₄ is the isogonal conjugate of X₃₂₇₃.

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

X₃₆₀₄ (ISOGONAL CONJUGATE OF X₃₂₇₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {356, 1134}, {357, 1137}

• Related Centers: X₃₂₇₃ (PERSPECTOR OF TRIANGLES M1M2M3 AND J1J2J3), X₃₅₆ (MORLEY CENTER), X₁₁₃₄ (3rd MORLEY-TAYLOR-MARR CENTER), X₃₅₇ (1st MORLEY-TAYLOR-MARR CENTER), X₁₁₃₇ (6th MORLEY-TAYLOR-MARR CENTER)

Trilinears: \(\csc(A/3 + \pi /3) : \csc(B/3 + \pi /3) : \csc(C/3 + \pi /3)\)

Barycentrics: \((\sin A)\csc(A/3 + \pi /3) : (\sin B)\csc(B/3 + \pi /3) : (\sin C)\csc(C/3 + \pi /3)\)

Search Value (6, 9, 13): \(1.754900100533\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines5 lines

{356, 1134}, {357, 1137}, {358, 1136}, {3273, 3603}, {3274, 3602}

▸ Functional Relations1 relation

• X₃₆₀₄ = trilinear pole of perspectrix of ABC and 3rd Morley triangle

Notes

X₃₆₀₄ is {X₁₁₃₄, X₁₁₃₅}-harmonic conjugate of X₃₂₇₇. (Bernard Gibert, November 3, 2010)

Let A'B'C' be the 1st Morley triangle. Let B_a and C_a be points on BC such that A'B_cC_a is an equilateral triangle; define C_b and A_c cyclically, and define A_b and B_c cyclically. Let A″ be the midpoint of A_b and A_c, and define B″ and C″ cyclically. Then A″B″C″, here named the 1st MORLEY-MIDPOINT TRIANGLE, is an equilateral triangle that is homothetic to A'B'C' and perspective to ABC, with perspector X₃₆₀₄. (Thanh Oai Dao and Peter Moses, March 30, 2018)

Let A'B'C' be the 3rd Morley triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₆₀₄. (Randy Hutson, July 11, 2019)

Let A'B'C' be the 3rd Morley triangle. Let A* be the foot of the perpendicular from A to line B'C', and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X₃₆₀₄. (Randy Hutson, July 11, 2019)