X₁₈₃

TRILINEAR PRODUCT X₇₅X_182

X₁₈₃ = X₆-of-circummedial-triangle

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

X₁₈₃ (TRILINEAR PRODUCT X₇₅X_182) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {3, 76}

• Related Centers: X₂₆₃ (ISOGONAL CONJUGATE OF X₁₈₃), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₃ (CIRCUMCENTER), X₇₆ (3rd BROCARD POINT)

Trilinears: \(b^{2}c^{2}\cos(A- \omega ) : c^{2}a^{2}\cos(B - \omega ) : a^{2}b^{2}\cos(C - \omega )\)

Barycentrics: \(\csc A \cos(A - \omega ) : \csc B \cos(B - \omega ) : \csc C \cos(C - \omega )\)

Search Value (6, 9, 13): \(5.092762853514\)

▸ Alternate Coordinates1 form

• Barycentrics: \(\cot A + \tan \omega : :\)

▸ Linear Combinations1 combination

• X₁₈₃ = \(3\) X₂ - \(2(\cos \omega )^{2}\) X₆

▸ Incident Central Lines15 lines

{2, 6}, {3, 76}, {5, 315}, {22, 157}, {25, 264}, {55, 350}, {95, 305}, {187, 1003}, {274, 474}, {316, 381}, {317, 427}, {383, 621}, {538, 574}, {622, 1080}, {668, 956}

▸ Functional Relations9 relations

• X₁₈₃ = isogonal conjugate of X₂₆₃

• X₁₈₃ = isotomic conjugate of X₂₆₂

• X₁₈₃ = X₆₄₅-beth conjugate of X₁₈₃

• X₁₈₃ = {X₂,X₆₉}-harmonic conjugate of X₃₂₅

• X₁₈₃ = X₆-of-circummedial-triangle

• X₁₈₃ = pole wrt circumcircle of trilinear polar of X₃₁₁₄ (line X₆₆₉X₈₀₄)

• X₁₈₃ = insimilicenter of Artzt and anti-Artzt circles; the exsimilicenter is X₂

• X₁₈₃ = crossdifference of every pair of points on the line through [U(2) of pedal triangle of P(1)] and [P(2) of pedal triangle of U(1)]

• X₁₈₃ = X₅₀₃₄-of-6th-Brocard-triangle

Notes

Let Cm_aCm_bCm_c be the circummedial triangle. Let L_a be the line through A parallel to Cm_bCm_c, and define L_b and L_c cyclically. Let A″ = L_b∩L_c, B″ = L_c∩L_a, C″ = L_a∩L_b. The lines Cm_aA″, Cm_bB″, Cm_cC″ concur in X₁₈₃. (Randy Hutson, December 26, 2015)

X₁₈₃ is the {X₂,X₆₉}-harmonic conjugate of X₃₂₅.