X₁₁₅₂

ISOGONAL CONJUGATE OF X₁₁₃₂

X₁₁₅₂ is the radical center of the Lucas(-1:1) circles and the perspector of triangle ABC and the Lucas(-1,1) inner triangle.

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

X₁₁₅₂ (ISOGONAL CONJUGATE OF X₁₁₃₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {2, 490}

• Related Centers: X₁₁₃₂ (ARCTAN(-2) KIEPERT POINT), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₂ (CENTROID), X₄₉₀ (CEVAPOINT OF X₂₀ AND X₄₈₈)

Trilinears: \(\sin A - 2 \cos A : :\)

Barycentrics: \(\sin^{2} A - \sin 2A : :\)

Search Value (6, 9, 13): \(-10.343109558395\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\sin(A - arctan(2)) : :\)

• Trilinears: \(a(2S_{A} - S) : :\)

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

▸ Incident Central Lines10 lines

{2, 490}, {3, 6}, {4, 615}, {30, 486}, {35, 1124}, {36, 1335}, {140, 485}, {141, 488}, {487, 524}, {590, 631}

▸ Functional Relations9 relations

• X₁₁₅₂ = isogonal conjugate of X₁₁₃₂

• X₁₁₅₂ = inverse-in-Brocard circle of X₁₁₅₁

• X₁₁₅₂ = X₄₉₄-Ceva conjugate of X₆

• X₁₁₅₂ = crosspoint of X₂₄₉ and X₁₃₀₇

• X₁₁₅₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (3, 372, 6), (3, 3312, 371), (371, 372, 3312), (371, 3312, 6)

• X₁₁₅₂ = exsimilicenter of circumcircle and Lucas(-1) inner circle; the insimilicenter is X₆₃₉₈

• X₁₁₅₂ = outer Soddy center (X₁₇₅) of tangential triangle, if ABC is acute

• X₁₁₅₂ = X₁-of-Lucas(-1)-central-triangle, if the Lucas(-1) circles are all externally tangent; otherwise, X₁₁₅₂ is an excenter of the Lucas(-1) central triangle

• X₁₁₅₂ = X₃-of-Lucas(-1)-tangents-triangle

Notes

X₁₁₅₂ is the perpsector of each pairs of the following five triangles: tangential triangle, Lucas(-1) tangents triangle, Lucas(-1) inner tangential triangle, 1st Lucas(-1) secondary tangents triangle, Lucas(-1) Brocard triangle. Also, X₁₁₅₂ perspector of Lucas(-8) central triangle and circumsymmedial triangle. (Randy Hutson, October 13, 2015)

Let A'B'C' be the Lucas(-1) tangents triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₁₅₂. (Randy Hutson, October 13, 2015)

Let A'B'C' be the Lucas(-1) central triangle. Let A″ be the pole, wrt the A-Lucas(-1) circle, of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₁₅₂. The pole A″ can also be constructed as the intersection of the polars of B' and C' wrt the A-Lucas(-1) circle, and similarly for B″ and C″. (Randy Hutson, October 13, 2015)

Let A'B'C' be the Lucas(-1) central triangle. Let A″ be the pole, wrt the A-Lucas(-1) circle, of line BC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₁₅₂. (Randy Hutson, October 13, 2015)

Let A'B'C' be the triangle whose trilinear vertex matrix is the sum of the matrices for the Lucas(-1) central and Lucas(-1) tangents triangles, so that A' =\(2a(S_{A} - S) : b(2SB - S) : c(2SC - S)\). The lines AA', BB', CC' concur in X₁₁₅₂. (Randy Hutson, September 14, 2016)

Let {a’} be the circle through B and C orthogonal to the circumcircle of ABC, and define {b’} and {c’} cyclically. The circle internally tangent to {a’}, {b’}, {c’} has center X₁₁₅₂; see X₁₁₅₁. (César Lozada, July 3, 2019)