X₁₃₆₈

COMPLEMENTARY CONJUGATE OF X₆

X₁₃₆₈ = midpoint of X₂₅ and X₁₃₇₀

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

X₁₃₆₈ (COMPLEMENTARY CONJUGATE OF X₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {25, 1370}, {2, 3} (Euler line)

• Related Centers: X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₁₃₇₀ (ANTICOMPLEMENT OF X₂₅), X₂ (CENTROID), X₃ (CIRCUMCENTER)

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

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

Search Value (6, 9, 13): \(4.217255557216\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines11 lines

{2, 3}, {11, 1040}, {12, 1038}, {98, 801}, {114, 122}, {120, 123}, {125, 343}, {126, 127}, {230, 577}, {495, 612}, {496, 614}

▸ Functional Relations14 relations

• X₁₃₆₈ = midpoint of X₂₅ and X₁₃₇₀

• X₁₃₆₈ = reflection of X₁₅₉₆ in X₅

• X₁₃₆₈ = complement of X₂₅

• X₁₃₆₈ = complementary conjugate of X₆

• X₁₃₆₈ = isotomic conjugate of isogonal conjugate of X₆₄₆₇

• X₁₃₆₈ = polar conjugate of isogonal conjugate of X₂₂₄₀₁

• X₁₃₆₈ = circumcircle-inverse of X₃₇₉₂₈

• X₁₃₆₈ = X_i-Ceva conjugate of X_j for these (i,j): (2, 1196), (670, 525), (222, 652), (255, 520)

• X₁₃₆₈ = crosspoint of X₂ and X₃₀₅

• X₁₃₆₈ = perspector of circumconic centered at X₁₁₉₆

• X₁₃₆₈ = center of circumconic that is locus of trilinear poles of lines passing through X₁₁₉₆

• X₁₃₆₈ = X₂-Ceva conjugate of X₁₁₉₆

• X₁₃₆₈ = homothetic center of the medial triangle and the 3rd pedal triangle of X₃

• X₁₃₆₈ = X₆₂₄₄-of-orthic-triangle if ABC is acute

Notes

As a point on the Euler line, X₁₃₆₈ has Shinagawa coefficients \((E - F, -E - F)\).