X₁₃₇₀

ANTICOMPLEMENT OF X₂₅

X₁₃₇₀ is the anticomplement of X₂₅.

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

X₁₃₇₀ (ANTICOMPLEMENT OF X₂₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {25, 1368}, {2, 3} (Euler line)

• Related Centers: X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₁₃₆₈ (COMPLEMENTARY CONJUGATE OF X₆), X₂ (CENTROID), X₃ (CIRCUMCENTER)

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

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = b^{2}/(a^{2} - b^{2} + c^{2}) + c^{2}/(a^{2} + b^{2} - c^{2}) - a^{2}/(- a^{2} + b^{2} + c^{2})\)

Search Value (6, 9, 13): \(8.980915851397\)

▸ Alternate Coordinates1 form

• Barycentrics: \(\tan B + \tan C - \tan A - \tan \omega : :\)

▸ Incident Central Lines5 lines

{2, 3}, {66, 69}, {305, 315}, {343, 1350}, {925, 1297}

▸ Functional Relations15 relations

• X₁₃₇₀ = reflection of X₂₅ in X₁₃₆₈

• X₁₃₇₀ = isogonal conjugate of X₃₄₂₀₇

• X₁₃₇₀ = isotomic conjugate of X₁₃₅₇₅

• X₁₃₇₀ = complement of X₇₅₀₀

• X₁₃₇₀ = anticomplement of X₂₅

• X₁₃₇₀ = crosspoint of X₆₇₀ and X₂₃₅₈₂

• X₁₃₇₀ = circumcircle-inverse of X₃₇₉₂₉

• X₁₃₇₀ = cevapoint of X_i and X_j for these (i,j): {159, 23115}, {455, 3162}

• X₁₃₇₀ = crosssum of X₆₆₉ and X₃₂₆₉

• X₁₃₇₀ = anticomplementary conjugate of X₁₉₃

• X₁₃₇₀ = X_i-Ceva conjugate of X_j for these (i,j): (305, 2), (315, 69)

• X₁₃₇₀ = homothetic center of anticomplementary triangle and 3rd antipedal triangle of X₄ (or polar triangle of anticomplementary circle)

• X₁₃₇₀ = pole, wrt polar circle, of the radical axis of any pair of {1st, 2nd and 3rd pedal circles of X₄}

• X₁₃₇₀ = pole of de Longchamps line wrt anticomplementary circle

• X₁₃₇₀ = inverse-in-de-Longchamps-circle of X₂₃

Notes

As a point on the Euler line, X₁₃₇₀ has Shinagawa coefficients \((E, -2E - 2F)\).