X₇₈

ISOGONAL CONJUGATE OF X₃₄

X₇₈ is the isogonal conjugate of X₃₄.

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

X₇₈ (ISOGONAL CONJUGATE OF X₃₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 2}, {9, 21}

• Related Centers: X₃₄ (-BETH CONJUGATE OF X₄), X₁ (INCENTER), X₂ (CENTROID), X₉ (MITTENPUNKT), X₂₁ (SCHIFFLER POINT)

Trilinears: \(1/(1 - \sec A) : 1/(1 - \sec B) : 1/(1 - \sec C)\)

Barycentrics: \(a/(1 - \sec A) : b/(1 - \sec B) : c/(1 - \sec C)\)

Search Value (6, 9, 13): \(5.339129461606\)

▸ Alternate Coordinates5 forms

• Trilinears: \(\cos A \csc^{2}(A/2) : :\)

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

• Trilinears: \(S_{A}(S_{A} + bc) : :\)

• Trilinears: \((b + c - a) \cot A : :\)

• Trilinears: \(\cot A \cot(A/2) : :\)

▸ Incident Central Lines29 lines

{1, 2}, {3, 63}, {4, 908}, {9, 21}, {20, 329}, {29, 33}, {37, 965}, {38, 988}, {40, 100}, {46, 758}, {55, 960}, {56, 480}, {57, 404}, {69, 73}, {101, 205}, {207, 653}, {210, 958}, {212, 283}, {220, 949}, {226, 377}, {271, 394}, {273, 322}, {280, 282}, {345, 1040}, {392, 1057}, {474, 942}, {517, 945}, {644, 728}, {999, 1059}

▸ Functional Relations13 relations

• X₇₈ = isogonal conjugate of X₃₄

• X₇₈ = isotomic conjugate of X₂₇₃

• X₇₈ = X_i-Ceva conjugate of X_j for these (i,j): (69, 63), (312, 9), (332, 345)

• X₇₈ = X_i-cross conjugate of X_j for these (i,j): (3, 271), (72, 8), (212, 9), (219, 63)

• X₇₈ = crosspoint of X₆₉ and X₃₄₅

• X₇₈ = crosssum of X_i and X_j for these (i,j): (25, 608), (56, 1406), (604, 1395), (1042, 1426)

• X₇₈ = X_i-beth conjugate of X_j for these (i,j): (78, 3), (643, 40), (1043, 1)

• X₇₈ = trilinear pole of line X₅₂₁X₆₅₂

• X₇₈ = {X₁,X₈}-harmonic conjugate of X₃₈₇₂

• X₇₈ = {X₂,X₁₄₅}-harmonic conjugate of X₉₃₈

• X₇₈ = X₉₂-isoconjugate of X₆₀₄

• X₇₈ = homothetic center of anticomplementary triangle and tangential triangle of the hexyl triangle

• X₇₈ = perspector of ABC and extraversion triangle of X₇₇