X₁₉₂

X₁-CEVA CONJUGATE OF X₂ (CONGRUENT PARALLELIANS POINT)

X₁₉₂ = anticomplement of X₇₅

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

X₁₉₂ (X₁-CEVA CONJUGATE OF X₂ (CONGRUENT PARALLELIANS POINT)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 37}, {1, 87}

• Related Centers: X₁ (INCENTER), X₂ (CENTROID), X₃₇ (CROSSPOINT OF INCENTER AND CENTROID), X₈₇ (-CROSS CONJUGATE OF X₁)

Trilinears: \(bc(ca + ab - bc) : ca(ab + bc - ca) : ab(bc + ca - ab)\)

Barycentrics: \(ca + ab - bc : ab + bc - ca : bc + ca - ab\)

Search Value (6, 9, 13): \(0.475187131173\)

▸ Incident Central Lines11 lines

{1, 87}, {2, 37}, {6, 190}, {7, 335}, {8, 256}, {9, 239}, {55, 385}, {69, 742}, {144, 145}, {315, 746}, {869, 1045}

▸ Functional Relations16 relations

• X₁₉₂ = reflection of X_i in X_j for these (i,j): (8, 984), (75, 37), (1278, 75)

• X₁₉₂ = isogonal conjugate of X₂₁₆₂

• X₁₉₂ = isotomic conjugate of X₃₃₀

• X₁₉₂ = complement of X₁₂₇₈

• X₁₉₂ = anticomplement of X₇₅

• X₁₉₂ = X₁-Ceva conjugate of X₂

• X₁₉₂ = crosspoint of X₁ and X₄₃

• X₁₉₂ = crosssum of X₁ and X₈₇

• X₁₉₂ = X₉-Hirst inverse of X₂₃₉

• X₁₉₂ = X₆₄₆-beth conjugate of X₁₉₂

• X₁₉₂ = perspector of anticomplementary triangle and Gemini triangle 4

• X₁₉₂ = perspector of Gemini triangles 4 and 6

• X₁₉₂ = X₁₉-isoconjugate of X₂₃₀₈₆

• X₁₉₂ = perspector of incentral triangle and inverse of n(Incentral)*n(Medial)

• X₁₉₂ = trilinear pole of line X₃₈₃₅X₄₀₈₃ (the perspectrix of ABC and Gemini triangle 16)

• X₁₉₂ = {X₃₇,X₇₅}-harmonic conjugate of X₂

Notes

The segments through X₁₉₂ parallel to the sidelines with endpoints on the sidelines have equal length. For references as early as 1881, see Hyacinthos message 2929 (Paul Yiu, May 29, 2001). See also

Sabrina Bier, “Equilateral Triangles Intercepted by Oriented Parallelians,” :cite:`Forum Geometricorum <forum-geometricorum>` 1 (2001) 25-32.