X₁₉₉₂

ORTHOCORRESPONDENT OF X₂

Let H_aH_bH_c be the orthic triangle. X₁₉₉₂ is the radical center of the 2nd Lemoine circles of triangles AH_bH_c, BH_cH_a, CH_aH_b. (Randy Hutson, July 31 2018)

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

X₁₉₉₂ (ORTHOCORRESPONDENT OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 193}

• Related Centers: X₂ (CENTROID), X₁₉₃ (-CEVA CONJUGATE OF X₂)

Trilinears: \(bc(b^{2} + c^{2} - 5a^{2}) : :\)

Barycentrics: \(b^{2} + c^{2} - 5a^{2} : :\)

Search Value (6, 9, 13): \(-0.643551802357\)

▸ Incident Central Lines9 lines

{2, 6}, {4, 542}, {30, 1351}, {145, 190}, {218, 1332}, {317, 1249}, {344, 1743}, {376, 511}, {575, 631}

▸ Functional Relations11 relations

• X₁₉₉₂ = midpoint of X₂ and X₁₉₃

• X₁₉₉₂ = reflection of X_i in X_j for these (i,j): (2, 6), (69, 2), (599, 597)

• X₁₉₉₂ = isogonal conjugate of X₂₁₄₄₈

• X₁₉₉₂ = isotomic conjugate of X₅₄₈₅

• X₁₉₉₂ = anticomplement of X₅₉₉

• X₁₉₉₂ = X₅₉₈-Ceva conjugate of X₂

• X₁₉₉₂ = perspector of ABC and unary cofactor triangle of 4th anti-Brocard triangle

• X₁₉₉₂ = X₄-of-anti-Artzt-triangle

• X₁₉₉₂ = {X₅₉₇,X₅₉₉}-harmonic conjugate of X₂

• X₁₉₉₂ = {X₃₇₇₈₅,X₃₇₇₈₆}-harmonic conjugate of X₂

• X₁₉₉₂ = trilinear pole of line X₁₄₉₉X₈₆₄₄ (the perspectrix of ABC and 1st Parry triangle, and the orthic axis of the Thomson triangle)

Notes

Let A'B'C' be the 1st Parry triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₉₉₂. (Randy Hutson, February 10, 2016)

Let A'B'C' be the anti-Artzt triangle. Let A″ be the barycentric product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₉₉₂. (Randy Hutson, December 10, 2016)