X₅₁₂

ISOGONAL CONJUGATE OF X₉₉

X₅₁₂ is the point in which the line of the 1st and 2nd Brocard points meets the line at infinity.

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

X₅₁₂ (ISOGONAL CONJUGATE OF X₉₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {110, 249}, {187, 237}

• Related Centers: X₉₉ (STEINER POINT), X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₂₄₉ (ISOGONAL CONJUGATE OF X₁₁₅), X₁₈₇ (INVERSE-IN-CIRCUMCIRCLE OF X), X₂₃₇ (-LINE CONJUGATE OF X₂)

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

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

▸ Alternate Coordinates2 forms

• Trilinears: \(\sin A (\cos 2B - \cos 2C) : :\)

• Barycentrics: \(S_{B}^{2} - S_{C}^{2} : :\)

▸ Linear Combinations1 combination

• X₅₁₂ = \(P(1) - U(1)\)

▸ Incident Central Lines59 lines

{1, 875}, {2, 3111}, {4, 879}, {6, 2444}, {25, 2433}, {30, 511}, {32, 878}, {39, 881}, {51, 1640}, {64, 2435}, {74, 842}, {98, 2698}, {99, 805}, {100, 2703}, {101, 2702}, {102, 2708}, {103, 2700}, {104, 2699}, {105, 2711}, {106, 2712}, {107, 2713}, {108, 2714}, {109, 2701}, {110, 249}, {111, 843}, {112, 2715}, {115, 2679}, {187, 237}, {263, 2395}, {316, 850}, {460, 2501}, {650, 2499}, {660, 1016}, {670, 886}, {764, 2650}, {884, 2440}, {1292, 2704}, {1293, 2705}, {1294, 2706}, {1295, 2707}, {1296, 2709}, {1297, 2710}, {1326, 2605}, {1491, 1734}, {1500, 2084}, {1570, 2451}, {1577, 2533}, {1691, 2483}, {1692, 3251}, {1968, 2909}, {2021, 2491}, {2024, 2507}, {2030, 2492}, {2031, 2510}, {2032, 2508}, {2142, 2143}, {2254, 2530}, {2378, 2379}, {2643, 3271}

▸ Functional Relations47 relations

• X₅₁₂ = isogonal conjugate of X₉₉

• X₅₁₂ = isotomic conjugate of X₆₇₀

• X₅₁₂ = anticomplementary conjugate of X₁₄₈

• X₅₁₂ = complementary conjugate of X₁₁₅

• X₅₁₂ = X_i-Ceva conjugate of X_j for these (i,j): (2, 1084), (4, 115), (66, 125), (99, 39), (110, 6), (112, 32), 1018,1500), (1306, 1504), (1307, 1505)

• X₅₁₂ = crosspoint of X_i and X_j for these (i,j): (4, 112), (6, 110), (83, 99)

• X₅₁₂ = crosssum of X_i and X_j for these (i,j): (1, 1019), (2, 523), (3, 525), (6, 669), (39, 512), (100, 190), (311, 850), (514, 1125), (643, 662)

• X₅₁₂ = orthopoint of X₅₁₁

• X₅₁₂ = vertex conjugate of X₁₅ and X₁₆

• X₅₁₂ = crossdifference of every pair of points on line X₂X₆

• X₅₁₂ = X₁₁₂-line conjugate of X₃₀

• X₅₁₂ = perspector of vertex-triangle and side triangle of circumcevian triangles of X₃ and X₆

• X₅₁₂ = bicentric difference of PU(1)

• X₅₁₂ = bicentric difference of PU(2)

• X₅₁₂ = ideal point of PU(1)

• X₅₁₂ = ideal point of PU(2)

• X₅₁₂ = ideal point of PU(26)

• X₅₁₂ = vertex conjugate of PU(2)

• X₅₁₂ = vertex conjugate of X₆ and X₁₈₇

• X₅₁₂ = trilinear pole of line X₃₅₁X₈₆₅ (line is tangent to Steiner inellipse at X₁₀₈₄)

• X₅₁₂ = perspector of hyperbola {A, B, C,X₂,X₆} (circumconic centered at X₁₀₈₄)

• X₅₁₂ = intercept of Lemoine axis and the line at infinity (trilinear polars of X₆ and X₂)

• X₅₁₂ = center of circumconic that is locus of trilinear poles of lines passing through X₁₀₈₄

• X₅₁₂ = perspector of ABC and the dual of the 1st Brocard triangle

• X₅₁₂ = X₅₁₂-of-2nd-Brocard triangle

• X₅₁₂ = X₆₉₀-of-orthocentroidal-triangle

• X₅₁₂ = X₆₉₀-of X₄-Brocard-triangle

• X₅₁₂ = orthic isogonal conjugate of X₁₁₅

• X₅₁₂ = incentral isogonal conjugate of X₁₁₅

• X₅₁₂ = incentral isotomic conjugate of X₂₆₄₃

• X₅₁₂ = X₆-isoconjugate of X₇₉₉

• X₅₁₂ = X₉₂-isoconjugate of X₄₅₅₈

• X₅₁₂ = exsimilicenter of antipedal circles of PU(1)

• X₅₁₂ = harmonic center of antipedal circles of PU(1)

• X₅₁₂ = trilinear pole of PU(105)

• X₅₁₂ = bicentric difference of PU(191)

• X₅₁₂ = ideal point of PU(191)

• X₅₁₂ = perspector of ABC and unary cofactor triangle of Steiner triangle

• X₅₁₂ = X₆₉₀ of 4th anti-Brocard triangle

• X₅₁₂ = Thomson-isogonal conjugate of X₉₈

• X₅₁₂ = Lucas-isogonal conjugate of X₉₈

• X₅₁₂ = Cundy-Parry Psi transform of X₁₄₂₆₅

• X₅₁₂ = barycentric product X₄*X₆₄₇

• X₅₁₂ = barycentric product of Jerabek hyperbola intercepts of orthic axis

• X₅₁₂ = barycentric product of Kiepert hyperbola intercepts of Lemoine axis

• X₅₁₂ = perspector of circlecevian triangles of PU(1)

• X₅₁₂ = orthic-isotomic conjugate of X₃₄₉₈₀

Notes

Let A'B'C' be the 1st Brocard triangle. Let L_a be the reflection of B'C' in the internal angle bisector of vertex angle A, and define L_b and L_c cyclically. Then the lines L_a, L_b, L_c are parallel, and they concur in X₅₁₂. (Randy Hutson, September 5, 2015)