X₃₆₀₃

ISOGONAL CONJUGATE OF X₃₂₇₅

X₃₆₀₃ is the isogonal conjugate of X₃₂₇₅.

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

X₃₆₀₃ (ISOGONAL CONJUGATE OF X₃₂₇₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {357, 1135}, {358, 1134}

• Related Centers: X₃₂₇₅ (PERSPECTOR OF TRIANGLES S1S2S3 AND J1J2J3), X₃₅₇ (1st MORLEY-TAYLOR-MARR CENTER), X₁₁₃₅ (4th MORLEY-TAYLOR-MARR CENTER), X₃₅₈ (2nd MORLEY-TAYLOR-MARR CENTER), X₁₁₃₄ (3rd MORLEY-TAYLOR-MARR CENTER)

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

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

Search Value (6, 9, 13): \(3.242159168691\)

▸ Incident Central Lines5 lines

{357, 1135}, {358, 1134}, {1136, 1137}, {3273, 3604}, {3275, 3602}

▸ Functional Relations2 relations

• X₃₆₀₃ = trilinear pole of perspectrix of ABC and 2nd Morley triangle

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

Notes

X₃₆₀₃ is {X₁₁₃₆, X₁₁₃₇}-harmonic conjugate of X₃₂₇₆. (Bernard Gibert, November 3, 2010).

Let A'B'C' be the 3rd 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 3rd MORLEY-MIDPOINT TRIANGLE, is an equilateral triangle that is homothetic to A'B'C' and perspective to ABC, with perspector X₃₆₀₃. (Thanh Oai Dao and Peter Moses, March 30, 2018)

Let A'B'C' be the 2nd 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 X₃₆₀₃. (Randy Hutson, July 11, 2019)

Let A'B'C' be the 2nd 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 X₃₆₀₃. (Randy Hutson, July 11, 2019)