X₃₂₅

X₂-HIRST INVERSE OF X₆₉

X₃₂₅ is the X₂-hirst inverse of X₆₉.

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

X₃₂₅ (X₂-HIRST INVERSE OF X₆₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {5, 76}

• Related Centers: X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₅ (NINE-POINT CENTER), X₇₆ (3rd BROCARD POINT)

Trilinears: \(\csc^{2} A : :\)

Barycentrics: \(\cot A - \tan \omega : :\)

Search Value (6, 9, 13): \(7.504301653149\)

▸ Alternate Coordinates2 forms

• Trilinears: \(bc(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2}) : :\)

• Barycentrics: \(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2} : :\)

X₃₂₅ lies on the Simson quartic Q101.

▸ Incident Central Lines75 lines

{2, 6}, {3, 315}, {4, 1975}, {5, 76}, {11, 350}, {12, 1909}, {20, 6337}, {22, 160}, {23, 3447}, {25, 317}, {30, 99}, {32, 3788}, {39, 626}, {51, 4121}, {74, 2855}, {75, 2886}, {95, 1799}, {98, 2065}, {100, 2856}, {110, 2857}, {111, 2858}, {114, 511}, {115, 538}, {140, 1078}, {147, 1503}, {187, 620}, {194, 5025}, {232, 297}, {250, 340}, {253, 6340}, {264, 305}, {274, 442}, {290, 3978}, {310, 3136}, {311, 1238}, {320, 1447}, {332, 2893}, {339, 1236}, {383, 622}, {441, 2966}, {523, 684}, {532, 6108}, {533, 6109}, {542, 5939}, {621, 1080}, {623, 6115}, {624, 6114}, {631, 3785}, {637, 6289}, {638, 6290}, {639, 3103}, {640, 3102}, {671, 5503}, {672, 4766}, {698, 1916}, {732, 2023}, {740, 5988}, {758, 5977}, {868, 2396}, {892, 1494}, {1235, 1594}, {1329, 6377}, {1368, 6375}, {1444, 4220}, {1506, 3934}, {2071, 5866}, {2482, 3849}, {3233, 6148}, {3734, 5475}, {3814, 6382}, {3829, 4479}, {3847, 4357}, {3932, 4518}, {5149, 5162}, {5965, 6036}

▸ Functional Relations23 relations

• X₃₂₅ = midpoint of X_i and X_j for these (i,j): (99, 316), (147, 5999), (298, 299), (5078, 5979)

• X₃₂₅ = reflection of X_i in X_j for these (i,j): (115, 625), (187, 620), (385, 230), (1513, 114), (2966, 441)

• X₃₂₅ = isogonal conjugate of X₁₉₇₆

• X₃₂₅ = isotomic conjugate of X₉₈

• X₃₂₅ = complement of X₃₈₅

• X₃₂₅ = anticomplement of X₂₃₀

• X₃₂₅ = orthoptic-circle-of-Steiner-circumellipe-inverse of X₃₈₉₄₀

• X₃₂₅ = orthoptic-circle-of-Steiner-inellipe-inverse of X₅₁₀₈

• X₃₂₅ = cevapoint of X₂ and X₁₄₇

• X₃₂₅ = X_i-cross conjugate of X_j for these (i,j): (114, 2), (511, 297)

• X₃₂₅ = crossdifference of every pair of points on line X₃₂X₅₁₂

• X₃₂₅ = X₂-Hirst inverse of X₆₉

• X₃₂₅ = {X₂,X₆₉}-harmonic conjugate of X₁₈₃

• X₃₂₅ = perspector of hyperbola {A, B, C,X₉₉,PU(37)}}

• X₃₂₅ = intersection of trilinear polars of X₉₉, P(37), and U(37)

• X₃₂₅ = crosspoint of X₂ and X₁₄₇ wrt both the excentral and anticomplementary triangles

• X₃₂₅ = trilinear pole of line X₂₇₉₉X₃₅₆₉

• X₃₂₅ = X₁₁₅-of-1st-anti-Brocard-triangle

• X₃₂₅ = X₁₁₄-of-anti-McCay-triangle

• X₃₂₅ = intersection of Simson line of X₉₉ (line X₁₁₄X₃₂₅) and trilinear polar of X₉₉ (line X₂X₆)

• X₃₂₅ = pole wrt polar circle of trilinear polar of X₆₅₃₁ (line X₂₅X₆₆₉)

• X₃₂₅ = X₄₈-isoconjugate (polar conjugate) of X₆₅₃₁

• X₃₂₅ = polar conjugate of isogonal conjugate of X₃₆₂₁₂

Notes

Let L_a be the line through A parallel to the Lemoine axis, and define L_b and L_c cyclically. Let M_a be the reflection of BC in L_a, and define M_b and M_c cyclically. Let A' = M_b∩M_c, and define B' and C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the Lemoine axis. The triangle A″B″C″ is homothetic to ABC, with center of homothety X₃₂₅; see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, November 30, 2015)