X₁₉₃

X₄-CEVA CONJUGATE OF X₂

X₁₉₃ = anticomplement of X₆₉

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

X₁₉₃ (X₄-CEVA CONJUGATE OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {23, 159}

• Related Centers: X₄ (ORTHOCENTER), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₂₃ (FAR-OUT POINT), X₁₅₉ (-OF-TANGENTIAL-TRIANGLE)

Trilinears: \((\csc A)(\cot B + \cot C - \cot A) : (\csc B)(\cot C + \cot A - \cot B) : (\csc C)(\cot A + \cot B - \cot C)\)

Barycentrics: \(\cot B + \cot C - \cot A : \cot C + \cot A - \cot B : \cot A + \cot B - \cot C\)

Search Value (6, 9, 13): \(-3.916472397203\)

▸ Alternate Coordinates2 forms

• Trilinears: \((S_{A} - a^{2})/a : (S_{B} - b^{2})/b : (S_{C} - c^{2})/c\)

• Barycentrics: \(3a^{2} - b^{2} - c^{2} : 3b^{2} - c^{2} - a^{2} : 3c^{2} - a^{2} - b^{2}\) (Milorad Stevanovic, 5/12/2003)

▸ Incident Central Lines17 lines

{2, 6}, {4, 1351}, {7, 239}, {8, 894}, {20, 185}, {23, 159}, {44, 344}, {66, 895}, {144, 145}, {146, 148}, {253, 287}, {317, 393}, {330, 959}, {371, 488}, {372, 487}, {608, 651}, {1839, 3187}

▸ Functional Relations20 relations

• X₁₉₃ = reflection of X_i in X_j for these (i,j): (3, 1353), (4, 1351), (69, 6), (1352, 576)

• X₁₉₃ = isogonal conjugate of X₈₇₇₀

• X₁₉₃ = isotomic conjugate of X₂₉₉₆

• X₁₉₃ = complement of X₂₀₀₈₀

• X₁₉₃ = anticomplement of X₆₉

• X₁₉₃ = anticomplementary conjugate of X₁₃₇₀

• X₁₉₃ = complementary conjugate of complement of X₃₆₆₁₆

• X₁₉₃ = X₄-Ceva conjugate of X₂

• X₁₉₃ = X₂-Hirst inverse of X₂₃₀

• X₁₉₃ = polar conjugate of X₃₄₂₀₈

• X₁₉₃ = X_i-beth conjugate of X_j for these (i,j): (645, 193), (662, 608)

• X₁₉₃ = perspector of pedal and antipedal triangles of X₄ (orthic and anticomplementary triangles)

• X₁₉₃ = perspector, wrt orthic triangle, of polar circle

• X₁₉₃ = anticomplementary isotomic conjugate of X₂₀

• X₁₉₃ = orthic-isogonal conjugate of X₂

• X₁₉₃ = trilinear pole of polar, wrt complement of polar circle, of X₆₉

• X₁₉₃ = pole of orthic axis wrt Steiner circumellipse

• X₁₉₃ = {X₃₈₅,X₇₇₇₄}-harmonic conjugate of X₂

• X₁₉₃ = endo-homothetic center of 3rd and 4th tri-squares central triangles

• X₁₉₃ = perspector, wrt anticomplementary triangle, of polar circle

Notes

Let A' be the reflection of the midpoint of segment BC in X₆, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₉₃. (Randy Hutson, 9/23/2011)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B', C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. A'B'C' is perspective to the orthic and anticomplementary triangles at X₁₉₃. (Randy Hutson, January 29, 2018)