X₅₂₃

ISOGONAL CONJUGATE OF X₁₁₀

As the isogonal conjugate of a point on the circumcircle, X₅₂₃ lies on the line at infinity.

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

X₅₂₃ (ISOGONAL CONJUGATE OF X₁₁₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {23, 385}, {230, 231}

• Related Centers: X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₂₃ (FAR-OUT POINT), X₃₈₅ (-HIRST INVERSE OF X₆), X₂₃₀ (-CEVA CONJUGATE OF X₁₁₄), X₂₃₁ (-CEVA CONJUGATE OF X₁₂₈)

Trilinears: \(\sin(B -C) : \sin(C - A) : \sin(A - B)\)

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

▸ Alternate Coordinates4 forms

• Barycentrics: \(\cot B \cos 2C - \cot C \cos 2B : :\)

• Barycentrics: \(S_{B} - S_{C} : :\)

• Barycentrics: \(\cot B - \cot C : :\)

• Barycentrics: \(\cos 2B - \cos 2C : :\)

X₅₂₃ lies on the Kiepert parabola.

▸ Incident Central Lines52 lines

{1, 2605}, {2, 1649}, {4, 1552}, {6, 879}, {11, 1090}, {12, 2599}, {23, 385}, {30, 511}, {59, 655}, {66, 2435}, {74, 477}, {75, 876}, {98, 842}, {99, 691}, {100, 1290}, {101, 2690}, {102, 2695}, {103, 2688}, {104, 2687}, {105, 2752}, {106, 2758}, {107, 1304}, {108, 2766}, {109, 2689}, {110, 476}, {111, 2770}, {112, 935}, {125, 2677}, {140, 1116}, {141, 882}, {160, 3164}, {230, 231}, {250, 648}, {253, 2419}, {325, 684}, {396, 3272}, {656, 2457}, {827, 1287}, {878, 3425}, {885, 2346}, {930, 1291}, {1086, 2643}, {1101, 2612}, {1222, 2403}, {1292, 2691}, {1293, 2692}, {1294, 2693}, {1295, 2694}, {1296, 2696}, {1297, 2697}, {2525, 2526}, {2594, 2616}

▸ Functional Relations50 relations

• X₅₂₃ = isogonal conjugate of X₁₁₀

• X₅₂₃ = isotomic conjugate of X₉₉

• X₅₂₃ = complementary conjugate of X₁₂₅

• X₅₂₃ = anticomplementary conjugate of X₃₄₄₈

• X₅₂₃ = X_i-Ceva conjugate of X_j for these (i,j): (1, 11), (2, 115), (4, 125), (99, 2), (100, 442), (107, 4), (108, 429), (110, 5), (112, 427), (254, 136), (264, 338), (476, 30), (685, 1503), (1113, 1312), (1114, 1313)

• X₅₂₃ = cevapoint of X₂ and X₁₄₈

• X₅₂₃ = cevapoint of polar circle intercepts of Euler line

• X₅₂₃ = X_i-cross conjugate of X_j for these (i,j): (115, 2), (125, 4)

• X₅₂₃ = crosspoint of X_i and X_j for these (i,j): (2, 99), (4, 107), (54, 110), (112, 251)

• X₅₂₃ = crossdifference of every pair of points on line X₃X₆

• X₅₂₃ = orthopoint of X₃₀

• X₅₂₃ = X₃₀-line conjugate of X₅₁₁

• X₅₂₃ = barycentric product of X₃₄₁₃ and X₃₄₁₄

• X₅₂₃ = crosssum of X_i and X_j for these (i,j): (3, 520), (5, 523), (6, 512), (101, 692), (141, 525), (184, 647), (215, 654), (513, 942), (521, 960), (924, 1147)

• X₅₂₃ = orthojoin of X₁₁₅

• X₅₂₃ = X_i-Hirst inverse of X_j for these (i,j): (6, 1316), (30, 542), (512, 804)

• X₅₂₃ = bicentric difference of PU(i) for these i: 4, 5, 11, 43, 45, 61, 132, 173

• X₅₂₃ = ideal point of PU(i) for these i: 4, 5, 11, 43, 45, 61, 132

• X₅₂₃ = crossdifference of PU(29)

• X₅₂₃ = trilinear pole of PU(40); see note above

• X₅₂₃ = barycentric product of PU(40)

• X₅₂₃ = trilinear product of PU(71)

• X₅₂₃ = perspector of the Kiepert hyperbola

• X₅₂₃ = intersection of the trilinear polars of any 2 points on the Kiepert hyperbola

• X₅₂₃ = intercept of orthic axis and the line at infinity (the trilinear polars of X₄ and X₂)

• X₅₂₃ = center of parabola {A, B, C,X₄₇₆,X₅₂₃}}, which is the locus of trilinear poles of lines passing through X₁₁₅

• X₅₂₃ = perspector of ABC and the side-triangle of the medial and orthic triangles

• X₅₂₃ = perspector of ABC and the vertex-triangle of the tangential triangles of the medial and Feuerbach triangles

• X₅₂₃ = X₅₂₆-of-orthocentroidal-triangle

• X₅₂₃ = X₅₂₆-of-X₄-Brocard-triangle

• X₅₂₃ = X₃₀-of-1st-Parry-triangle

• X₅₂₃ = X₃₀-of-2nd-Parry-triangle

• X₅₂₃ = orthic isogonal conjugate of X₁₂₅

• X₅₂₃ = incentral isogonal conjugate of X₃₀₂₄

• X₅₂₃ = pole wrt polar circle of the Euler line

• X₅₂₃ = X₄₈-isoconjugate (polar conjugate) of X₆₄₈

• X₅₂₃ = X₆-isoconjugate of X₆₆₂

• X₅₂₃ = X₁₁₀₁-isoconjugate of X₅₂₃

• X₅₂₃ = X₂-vertex conjugate of X₂₃

• X₅₂₃ = X₃-vertex conjugate of X₃₀

• X₅₂₃ = X₄-vertex conjugate of X₁₈₆

• X₅₂₃ = barycentric cube root of X₈₀₂₉

• X₅₂₃ = perspector of side- and vertex-triangles of circumanticevian triangles of X₂ and X₄

• X₅₂₃ = barycentric square root of X₁₁₅

• X₅₂₃ = Cundy-Parry Psi transform of X₁₄₂₆₄

• X₅₂₃ = Thomson isogonal conjugate of X₇₄

• X₅₂₃ = Lucas isogonal conjugate of X₇₄

• X₅₂₃ = trilinear square root of X₁₁₀₉

• X₅₂₃ = barycentric product X₁₀*X₅₁₄

• X₅₂₃ = trilinear product of Kiepert hyperbola intercepts of antiorthic axis

Notes

Let A'B'C' be the 1st Brocard triangle. Let A″B″C″ be the 2nd Brocard triangle. Let A* be the trilinear pole of line A'A″, and define B* and C* cyclically. The lines AA*, BB*, CC* are parallel and meet the line at infinity at X₅₂₃. (Randy Hutson, December 26, 2015)

Let Eu_aEu_bEu_c be the Feuerbach triangle. Let L_a be the tangent to conic {A, B, C,Eu_b,Eu_c}} at A, and 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 Eu_aA″, Eu_bB″, Eu_cC″ concur in X₅₂₃; see also X₁₁₀₉.

Let L be the line X₁₁₅X₁₂₅; then X₅₂₃ = trilinear pole of L. The line L is also PU(40), the tangent to Steiner inellipse at X₁₁₅, the Lemoine axis of the 4th Brocard triangle, the Fermat axis of the 2nd Parry triangle, and the isogonal conjugate of hyperbola {A, B, C,PU(2)}) (Randy Hutson, December 26, 2015)