X₁₁₅₁

ISOGONAL CONJUGATE OF X₁₁₃₁

X₁₁₅₁ is the radical center of the Lucas circles, the incenter of the Lucas central triangle, and the perspector of triangle ABC and the Lucas inner triangle.

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

X₁₁₅₁ (ISOGONAL CONJUGATE OF X₁₁₃₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {2, 489}

• Related Centers: X₁₁₃₁ (ARCTAN(2) KIEPERT POINT), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₂ (CENTROID), X₄₈₉ (CEVAPOINT OF X₂₀ AND X₄₈₇)

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

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

Search Value (6, 9, 13): \(3.298808798294\)

▸ Alternate Coordinates3 forms

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

• Trilinears: \(a(2 S_{A} + S) : :\)

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

▸ Linear Combinations3 combinations

• X₁₁₅₁ = \(La/R_{a} + Lb/R_{b} + Lc/R_{c} +\) X₃ / \(R\), where L_a, L_b, L_c are the centers of the Lucas circles, and \(R_{a}\), \(R_{b}\), \(R_{c}\) their radii

• X₁₁₅₁ = \(La/R_{a} + Lb/R_{b} + Lc/R_{c} - Li/Ri\), where Li, Ri are the center and radius of the Lucas inner circle

• X₁₁₅₁ = \((R_{b}+R_{c}) La + (R_{c}+R_{a}) Lb + (R_{a}+R_{b}) Lc\) (Combos by Randy Hutson, September 5, 2015)

▸ Incident Central Lines10 lines

{2, 489}, {3, 6}, {4, 590}, {30, 485}, {35, 1335}, {36, 1124}, {140, 486}, {141, 487}, {488, 524}, {615, 631}

▸ Functional Relations11 relations

• X₁₁₅₁ = reflection of X₄₈₅ in X₈₉₈₁

• X₁₁₅₁ = isogonal conjugate of X₁₁₃₁

• X₁₁₅₁ = inverse-in-Brocard circle of X₁₁₅₂

• X₁₁₅₁ = X₄₉₃-Ceva conjugate of X₆

• X₁₁₅₁ = crosspoint of X₂₄₉ and X₁₃₀₆

• X₁₁₅₁ = perspector of Lucas(8) central triangle and circumsymmedial triangle

• X₁₁₅₁ = insimilicenter of circumcircle and Lucas inner circle

• X₁₁₅₁ = inner Soddy center (X₁₇₆) of tangential triangle, if ABC is acute

• X₁₁₅₁ = X₃-of-Lucas-tangents-triangle

• X₁₁₅₁ = X₂₀-of-X₂-quadsquares-triangle

• X₁₁₅₁ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (3, 371, 6), (3, 3311, 372), (371, 372, 3311), (372, 3311, 6)

Notes

Fourteen constructions for X₁₁₅₁ received from Randy Hutson, September 5, 2015:

(1)-(10): Each pair of the following triangles are perspective, and their perspector is X₁₁₅₁: tangential triangle, Lucas tangents triangle, Lucas inner tangential triangle, 1st Lucas secondary tangents triangle, Lucas Brocard triangle.

11. Let A', B', C' be the centers of the Kenmotu squares. Let A″ be the reflection of A' in X₃₇₁, and define B″ and C″ cyclically. The triangle A″B″C″ is homothetic to ABC at X₁₁₅₁.

12. Let A'B'C' be the Lucas tangents triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₁₅₁.

13. Let A'B'C' be the Lucas central triangle. Let A″ be the pole, wrt the A-Lucas circle, of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₁₅₁. The point A″ is also the intersection of the polars of B' and C' wrt the A-Lucas circle, and likewise for B″ and C″.

14. Let A'B'C' be the Lucas central triangle. Let A″ be the pole, wrt the A-Lucas circle, of line BC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₁₅₁.

Let A'B'C' be the triangle whose trilinear vertex matrix is the sum of the matrices for the Lucas central and Lucas tangents triangles, so that A' =\(2a(S_{A} + S) : b(2SB + S) : c(2SC + S)\). The lines AA', BB', CC' concur in X₁₁₅₁. (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 externally tangent to {a’}, {b’}, {c’} has center X₁₁₅₁; see X₁₁₅₂. (César Lozada, July 3, 2019)

X₁₁₅₁ is the {X₃,X₆}-harmonic conjugate of X₁₁₅₂.