X₂₁₈₅

X₂-ISOCONJUGATE OF X₁₈₁

X₂₁₈₅ = isogonal conjugate of X₂₁₇₁

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

X₂₁₈₅ (X₂-ISOCONJUGATE OF X₁₈₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {21, 60}, {27, 86}

• Related Centers: X₂₁₇₁ (-ISOCONJUGATE OF X₆₀), X₂₁ (SCHIFFLER POINT), X₆₀ (ISOGONAL CONJUGATE OF X₁₂), X₂₇ (CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER), X₈₆ (CEVAPOINT OF INCENTER AND CENTROID)

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

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

Search Value (6, 9, 13): \(2.777973913915\)

▸ Incident Central Lines13 lines

{1, 849}, {2, 662}, {21, 60}, {27, 86}, {55, 643}, {81, 593}, {110, 1621}, {171, 1326}, {222, 1414}, {261, 284}, {312, 2268}, {501, 1125}, {552, 553}

▸ Functional Relations10 relations

• X₂₁₈₅ = isogonal conjugate of X₂₁₇₁

• X₂₁₈₅ = isotomic conjugate of X₆₃₅₈

• X₂₁₈₅ = complement of anticomplementary conjugate of X₂₁₂₇₃

• X₂₁₈₅ = anticomplement of complementary conjugate of X₂₁₂₃₃

• X₂₁₈₅ = X₆-isoconjugate of X₁₂

• X₂₁₈₅ = X_i-Ceva conjugate of X_j for these (i,j): (249, 662), (261, 1098), (1509, 757)

• X₂₁₈₅ = cevapoint of X_i and X_j for these (i,j): (1, 572), (21, 284), (81, 1790)

• X₂₁₈₅ = X_i-cross conjugate of X_j for these (i,j): (21, 261), (60, 757), (284, 60), (2150, 270)

• X₂₁₈₅ = crosspoint of X₂₆₁ and X₁₅₀₉

• X₂₁₈₅ = crosssum of X₁₈₁ and X₁₅₀₀

Notes

Let DEF be the cevian triangle of the incenter. Let O_A be the circumcircle of AEF. Let T₁ be the line tangent to O_A at E, and let T₂ be the line tangent to O_A at F. Let A_B = T₁∩BC and A_C = T₂∩BC, and let A' be the radical center of the circumcircles of AEF, BFA_B, CAA_C.. Define B' and C' cyclically. The lines AA', BB', CC' concur in X₂₁₈₅. (Angel Montesdeoca, January 4, 2021)