X₆₉₄

ISOGONAL CONJUGATE OF X₃₈₅

X₆₉₄ is the isogonal conjugate of X₃₈₅.

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

X₆₉₄ (ISOGONAL CONJUGATE OF X₃₈₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {37, 256}, {110, 251}

• Related Centers: X₃₈₅ (-HIRST INVERSE OF X₆), X₃₇ (CROSSPOINT OF INCENTER AND CENTROID), X₂₅₆ (1st SHARYGIN POINT), X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₂₅₁ (ISOGONAL CONJUGATE OF X₁₄₁)

Trilinears: \(a/(a^{4} - b^{2}c^{2}) : :\)

Search Value (6, 9, 13): \(-0.132071240751\)

▸ Incident Central Lines11 lines

{6, 1084}, {37, 256}, {42, 893}, {110, 251}, {111, 805}, {141, 308}, {172, 904}, {257, 335}, {351, 881}, {384, 695}, {882, 888}

▸ Functional Relations24 relations

• X₆₉₄ = reflection of X_i in X_j for these (i,j): (6, 1084), (670, 141)

• X₆₉₄ = isogonal conjugate of X₃₈₅

• X₆₉₄ = isotomic conjugate of X₃₉₇₈

• X₆₉₄ = anticomplement of X₃₉₀₈₀

• X₆₉₄ = crossdifference of PU(185)

• X₆₉₄ = perspector of conic {A, B, C,X₈₀₅,X₁₈₈₂₉,X₃₆₈₉₇}}

• X₆₉₄ = X₆₃-isoconjugate of X₄₁₉

• X₆₉₄ = cevapoint of X₃₈₄ and X₃₈₅

• X₆₉₄ = X_i-cross conjugate of X_j for these (i,j): (446, 232), (511, 6)

• X₆₉₄ = trilinear pole of PU(1) (line X₃₉X₅₁₂)

• X₆₉₄ = barycentric product of circumcircle intercepts of line PU(11), alias X₁₄₁X₅₂₃)

• X₆₉₄ = antipode of X₆ in hyperbola {A, B, C,X₂,X₆}}

• X₆₉₄ = perspector of circumconic centered at X₁₅₆₇

• X₆₉₄ = center of circumconic that is locus of trilinear poles of lines passing through X₁₅₆₇

• X₆₉₄ = X₂-Ceva conjugate of X₃₉₀₉₂

• X₆₉₄ = cevapoint of X₆ and X₂₀₇₆

• X₆₉₄ = polar conjugate of isotomic conjugate of X₃₆₂₁₄

• X₆₉₄ = perspector of ABC and unary cofactor triangle of 1st Brocard triangle

• X₆₉₄ = perspector of ABC and unary cofactor triangle of 3rd Brocard triangle

• X₆₉₄ = perspector of ABC and unary cofactor triangle of 1st Neuberg triangle

• X₆₉₄ = center of the perspeconic of these triangles: 1st and 3rd Brocard

• X₆₉₄ = perspector of ABC and vertex-triangle of anticevian triangles of PU(1)

• X₆₉₄ = center of bicevian conic of PU(6)

• X₆₉₄ = barycentric product of Steiner circumellipse intercepts of line PU(1)

Notes

Let A'B'C' be the 1st Brocard triangle. Let L_a be the trilinear polar of A', and define L_b and L_c cyclically. Let A″ = L_b∩L_c, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₆₉₄. (Randy Hutson, December 26, 2015)

Let A'B'C' be the 1st Brocard triangle. Let L_a be the trilinear polar, wrt A'B'C', of A, and define L_b, L_c cyclically. Let A″ = L_b∩L_c, and define B″, C″ cyclically. A'A″, B'B″, C'C″ concur in X₆₉₄. (Randy Hutson, December 26, 2015)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₈₅. Let A* be the trilinear pole of line B″C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₆₉₄. (Randy Hutson, December 26, 2015)

Let DEF and D'E'F' be the 1st and 2nd Sharygin triangles. Let A' be the barycentric product D*D', and define B' and C' cyclically. The lines AA', BB', CC' concur in X₆₉₄. (Randy Hutson, December 26, 2015)