X₈₉₅

ISOGONAL CONJUGATE OF X₄₆₈

X₈₉₅ is the isogonal conjugate of X₄₆₈.

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

X₈₉₅ (ISOGONAL CONJUGATE OF X₄₆₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {193, 3448}

• Related Centers: X₁₉₃ (-CEVA CONJUGATE OF X₂), X₃₄₄₈ (ISOGONAL CONJUGATE OF X₃₄₄₇)

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

Search Value (6, 9, 13): \(1.078254016030\)

X₈₉₅ lies on the MacBeath circumconic, the Darboux septic.

▸ Incident Central Lines10 lines

{4, 542}, {6, 110}, {54, 575}, {65, 651}, {66, 193}, {67, 524}, {69, 125}, {74, 511}, {287, 879}, {290, 892}

▸ Functional Relations15 relations

• X₈₉₅ = midpoint of X₁₉₃ and X₃₄₄₈

• X₈₉₅ = reflection of X_i in X_j for these (i,j): (69, 125), (110, 6)

• X₈₉₅ = isogonal conjugate of X₄₆₈

• X₈₉₅ = anticomplement of X₅₁₈₁

• X₈₉₅ = circumcircle-inverse of X₆₀₉₁

• X₈₉₅ = trilinear pole of line X₃X₆₄₇

• X₈₉₅ = antigonal image of X₆₉

• X₈₉₅ = syngonal conjugate of X₆

• X₈₉₅ = polar conjugate of X₃₇₇₇₈

• X₈₉₅ = pole wrt polar circle of trilinear polar of X₃₇₇₇₈ (line X₆₉₀X₁₂₈₂₈)

• X₈₉₅ = MacBeath-circumconic-antipode of X₁₁₀

• X₈₉₅ = X₉₂-isoconjugate of X₁₈₇

• X₈₉₅ = perspector of ABC and unary cofactor triangle of 4th Brocard triangle

• X₈₉₅ = eigencenter of 2nd Ehrmann triangle

• X₈₉₅ = perspector of 2nd Ehrmann triangle and cross-triangle of ABC and 2nd Ehrmann triangle

Notes

Let A″B″C″ be the 2nd Ehrmann triangle. Let A* be the cevapoint of B″ and C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₈₉₅. (Randy Hutson, November 18, 2015)