X₆₄

ISOGONAL CONJUGATE OF X₂₀

X₆₄ is the isogonal conjugate of X₂₀.

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

X₆₄ (ISOGONAL CONJUGATE OF X₂₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1498, 3}

• Related Centers: X₁₄₉₈ (REFLECTION OF X₆₄ IN X₃), X₃ (CIRCUMCENTER)

Trilinears: \(1/(\cos A - \cos B \cos C) : 1/(\cos B - \cos C \cos A) : 1/(\cos C - \cos A \cos B)\)

Barycentrics: \(a/(\cos A - \cos B \cos C) : b/(\cos B - \cos C \cos A) : c/(\cos C - \cos A \cos B)\)

Search Value (6, 9, 13): \(31.627324066040\)

▸ Alternate Coordinates1 form

• Barycentrics: \(a^{2}/[3a^{4} - 2a^{2}(b^{2} + c^{2}) - (b^{2} - c^{2})^{2}] : :\)

X₆₄ lies on the Darboux cubic.

▸ Incident Central Lines16 lines

{1, 3182}, {3, 154}, {4, 3183}, {6, 185}, {20, 69}, {24, 74}, {30, 68}, {33, 65}, {40, 72}, {54, 378}, {55, 73}, {71, 198}, {84, 3353}, {265, 382}, {3345, 3472}, {3346, 3355}

▸ Functional Relations24 relations

• X₆₄ = reflection of X₁₄₉₈ in X₃

• X₆₄ = isogonal conjugate of X₂₀

• X₆₄ = isotomic conjugate of X₁₄₆₁₅

• X₆₄ = complement of X₆₂₂₅

• X₆₄ = anticomplement of X₂₈₈₃

• X₆₄ = circumcircle-inverse of X₁₁₅₈₉

• X₆₄ = X₂₅-cross conjugate of X₆

• X₆₄ = X₁-beth conjugate of X₂₀₇

• X₆₄ = crosspoint of X₄ and X₃₃₄₆

• X₆₄ = crosssum of X₃ and X₁₄₉₈

• X₆₄ = perspector of hexyl triangle and anticevian triangle of X₂₁₈₄

• X₆₄ = trilinear pole of line X₆₄₇X₆₅₇

• X₆₄ = concurrence of normals to MacBeath circumconic at A, B, C

• X₆₄ = isogonal conjugate, wrt tangential triangle of MacBeath circumconic (or anticevian triangle of X₃), of X₁₄₉₈

• X₆₄ = orthocenter of x(3)X₆X₂₄₃₅

• X₆₄ = orthology center of ABC and half-altitude triangle

• X₆₄ = intersection of tangents at X₃ and X₄ to Thomson cubic K002

• X₆₄ = intersection of tangents at X₂₀ and X₆₄ to Darboux cubic K004

• X₆₄ = perspector of ABC and the reflection in X₃ of the antipedal triangle of X₃ (tangential triangle)

• X₆₄ = perspector of ABC and circumcircle antipode of circumanticevian triangle of X₃

• X₆₄ = perspector of ABC and unary cofactor triangle of half-altitude triangle

• X₆₄ = X₂₁₃₆-of-orthic-triangle if ABC is acute

• X₆₄ = X₈₉₀₅-of-excentral-triangle

• X₆₄ = X₃-vertex conjugate of X₃

Notes

A construction of X₆₄ appears in Lemoine’s 1886 paper cited at X₁₉.

Let A'B'C' be the half-altitude 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″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₆₄. (Randy Hutson, November 18, 2015)

Let O_a be the circle with segment BC as diameter. Let A' be the perspector of O_a. Let L_a be the polar of A' wrt O_a. 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 AA″, BB″, CC″ concur in X₆₄. (Randy Hutson, November 18, 2015)

Let A'B'C' be the cevian triangle of X₆₉. Let A″ be the orthocenter of AB'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₆₄. (Randy Hutson, November 18, 2015)

Let A'B'C' be the reflection of the orthic triangle in X₄. Let A″B″C″ be the trangential triangle, with respect ot the orthic triangle, of the circumconic of the orthic triangle with center X₄; i.e., the bicevian conic of X₄ and X₄₅₉. Then X₆₄ is the perspector of A'B'C' and A″B″C″. (Randy Hutson, November 18, 2015)

The tangents at A, B, C to the Darboux cubic K004 concur in X₆₄. (Randy Hutson, November 18, 2015)