X₁₉₄

X₆-CEVA CONJUGATE OF X₂

X₁₉₄ = reflection of X₇₆ in X₃₉

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

X₁₉₄ (X₆-CEVA CONJUGATE OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {76, 39}

• Related Centers: X₇₆ (3rd BROCARD POINT), X₃₉ (BROCARD MIDPOINT)

Trilinears: \(bc[a^{2}b^{2} + a^{2}c^{2} - b^{2}c^{2}] : :\)

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

Search Value (6, 9, 13): \(-1.630187791614\)

▸ Alternate Coordinates1 form

• Barycentrics: \(\cot^{2} A - \csc^{2} A \cos 2\omega : :\) (M. Iliev, 5/13/07)

▸ Linear Combinations1 combination

• X₁₉₄ = \(2\) X₃₉ - X₇₆ - \(P(1) + U(1) -\) X₇₆

▸ Incident Central Lines50 lines

{1, 87}, {2, 39}, {3, 385}, {4, 147}, {6, 384}, {8, 730}, {10, 3097}, {20, 185}, {32, 99}, {63, 239}, {69, 695}, {75, 1107}, {83, 3734}, {183, 5013}, {184, 3492}, {190, 2176}, {257, 986}, {262, 2996}, {263, 3498}, {304, 3797}, {315, 736}, {325, 5025}, {350, 2275}, {401, 1993}, {487, 1587}, {488, 1588}, {548, 3793}, {574, 1078}, {616, 3104}, {617, 3105}, {627, 3106}, {628, 3107}, {648, 1968}, {712, 4393}, {1007, 2023}, {1593, 1941}, {1654, 4201}, {1670, 2547}, {1671, 2546}, {1909, 2276}, {2128, 2285}, {3096, 4045}, {3212, 3503}, {3314, 3933}, {3413, 3557}, {3414, 3558}, {3522, 5188}, {3770, 4261}, {3906, 5652}, {3972, 5007}

▸ Functional Relations26 relations

• X₁₉₄ = reflection of X₇₆ in X₃₉

• X₁₉₄ = isogonal conjugate of X₃₂₂₄

• X₁₉₄ = isotomic conjugate of X₂₉₉₈

• X₁₉₄ = isotomic conjugate of the complement of X₃₂₇₄₇

• X₁₉₄ = complement of X₂₀₀₈₁

• X₁₉₄ = anticomplement of X₇₆

• X₁₉₄ = circumcircle-inverse of X₃₂₅₁₇

• X₁₉₄ = polar circle inverse of X₃₂₅₂₇

• X₁₉₄ = anticomplementary conjugate of X₃₁₅

• X₁₉₄ = complementary conjugate of complement of X₃₆₆₁₅

• X₁₉₄ = eigencenter of cevian triangle of X₆

• X₁₉₄ = eigencenter of anticevian triangle of X₂

• X₁₉₄ = radical center of the Neuberg circles.

• X₁₉₄ = X₆-Ceva conjugate of X₂

• X₁₉₄ = X₃-Hirst inverse of X₃₈₅

• X₁₉₄ = anticomplementary-isotomic conjugate of X₆₉

• X₁₉₄ = X₆₃₇₄-cross conjugate of X₂

• X₁₉₄ = vertex conjugate of PU(140)

• X₁₉₄ = 1st-Brocard-to-6th-Brocard similarity image of X₆

• X₁₉₄ = X₉₉-of-6th-Brocard-triangle

• X₁₉₄ = anticomplementary-circle-inverse of X₃₂₅₂₈

• X₁₉₄ = orthoptic-circle-of-Steiner-inellipse-inverse of X₃₂₅₃₀

• X₁₉₄ = orthoptic-circle-of-Steiner-circumellipe-inverse of X₃₂₅₂₆

• X₁₉₄ = de Longchamps-circle-inverse of X₃₂₅₂₉

• X₁₉₄ = 2nd-Brocard-circle-inverse of X₃₂₅₃₁

• X₁₉₄ = perspector of 3rd Brocard triangle and 1st Brocard-reflected triangle

Notes

Let O_a be the circle through A and tangent to BC at its midpoint. Define O_b and O_c cyclically. The radical center of O_a, O_b, O_c is X₁₉₄. (Randy Hutson, December 26, 2015)

Let A_aB_aC_a, A_bB_bC_b, A_cB_cC_c be the A-, B- and C-anti-altimedial triangles. Let A' be the trilinear product A_a*A_b*A_c, and define B', C' cyclically. Triangle A'B'C' is the anticomplementary triangle of the 1st Brocard triangle, and is perspective to ABC at X₄, and to the anticomplementary triangle at X₁₉₄. (Randy Hutson, November 2, 2017)

Let O_A be the circle centered at the A-vertex of the 1st Neuberg triangle and passing through A; define O_B and O_C cyclically. X₁₉₄ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

X₁₉₄ is the {X₃₉,X₇₆}-harmonic conjugate of X₂.