X₈₂

ISOGONAL CONJUGATE OF X₃₈

X₈₂ is the isogonal conjugate of X₃₈.

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

X₈₂ (ISOGONAL CONJUGATE OF X₃₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {31, 75}, {10, 83}

• Related Centers: X₃₈ (CROSSPOINT OF X₁ AND X₇₅), X₃₁ (2nd POWER POINT), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER), X₁₀ (SPIEKER CENTER), X₈₃ (CEVAPOINT OF CENTROID AND SYMMEDIAN POINT)

Trilinears: \(1/(b^{2} + c^{2}) : 1/(c^{2} + a^{2}) : 1/(a^{2} + b^{2})\)

Barycentrics: \(a/(b^{2} + c^{2}) : b/(c^{2} + a^{2}) : c/(a^{2} + b^{2})\)

Search Value (6, 9, 13): \(1.057543146404\)

▸ Alternate Coordinates1 form

• Trilinears: \(\sin A \csc(A + \omega ) : \sin B \csc(B + \omega ) : \sin C \csc(C + \omega )\)

▸ Incident Central Lines7 lines

{1, 560}, {10, 83}, {31, 75}, {37, 251}, {58, 596}, {689, 715}, {759, 827}

▸ Functional Relations8 relations

• X₈₂ = isogonal conjugate of X₃₈

• X₈₂ = isotomic conjugate of X₁₉₃₀

• X₈₂ = anticomplement of X₂₁₂₄₉

• X₈₂ = cevapoint of X₁ and X₃₁

• X₈₂ = trilinear pole of line X₆₆₁X₈₃₀

• X₈₂ = crossdifference of every pair of points on line X₂₀₈₄X₂₅₃₀

• X₈₂ = perspector of ABC and extraversion triangle of X₈₂ (which is also the anticevian triangle of X₈₂)

• X₈₂ = crosspoint of X₁ and X₃₁ wrt the excentral triangle

Notes

Let Cm_aCm_bCm_c be the circummedial triangle. Let A″ be the trilinear product Cm_b*Cm_c, and define B″, C″ cyclically. A″, B″, C″ are collinear on line X₇₉₈X₈₁₂ (the trilinear polar of X₃₁₁₂). The lines AA″, BB″, CC″ concur in X₈₂. (Randy Hutson, October 15, 2018)