X₇₁

ISOGONAL CONJUGATE OF X₂₇

X₇₁ is the isogonal conjugate of X₂₇.

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

X₇₁ (ISOGONAL CONJUGATE OF X₂₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 9}, {6, 31}

• Related Centers: X₂₇ (CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER), X₄ (ORTHOCENTER), X₉ (MITTENPUNKT), X₆ (SYMMEDIAN POINT), X₃₁ (2nd POWER POINT)

Trilinears: \((b + c) \cos A : (c + a) \cos B : (a + b) \cos C\)

Barycentrics: \((b + c) \sin 2A : (c + a) \sin 2B : (a + b) \sin 2C\)

Search Value (6, 9, 13): \(5.850835756503\)

▸ Incident Central Lines13 lines

{1, 579}, {3, 48}, {4, 9}, {6, 31}, {35, 284}, {37, 65}, {54, 572}, {63, 69}, {64, 198}, {74, 101}, {165, 610}, {190, 290}, {583, 1100}

▸ Functional Relations15 relations

• X₇₁ = isogonal conjugate of X₂₇

• X₇₁ = anticomplement of X₃₄₈₃₀

• X₇₁ = X_i-Ceva conjugate of X_j for these (i,j): (3, 228), (9, 37), (10, 42), (63, 72)

• X₇₁ = X₂₂₈-cross conjugate of X₇₃

• X₇₁ = crosspoint of X_i and X_j for these (i,j): (3, 63), (9, 219), (10, 306)

• X₇₁ = crosssum of X_i and X_j for these (i,j): (1, 579), (4, 19), (28, 1127), (57, 278), (58, 1474)

• X₇₁ = crossdifference of every pair of points on line X₂₄₂X₅₁₄

• X₇₁ = X₄-line conjugate of X₂₄₂

• X₇₁ = X_i-beth conjugate of X_j for these (i,j): (219, 71), (1018, 71)

• X₇₁ = trilinear pole of line X₆₄₇X₈₁₀

• X₇₁ = X₉₂-isoconjugate of X₅₈

• X₇₁ = barycentric product of Jerabek hyperbola intercepts of Nagel line

• X₇₁ = polar conjugate of isotomic conjugate of X₃₆₈₂

• X₇₁ = antigonal conjugate of X₃₈₅₃₅

• X₇₁ = X₆₃-isoconjugate of X₈₇₄₇

Notes

X₇₁ is the intersection of the isotomic conjugate of the polar conjugate of the Nagel line (i.e., line X₆₃X₆₉), and the polar conjugate of the isotomic conjugate of the Nagel line (i.e., line X₄X₉). (Randy Hutson, July 11, 2019)

X₇₁ is the {X₉,X₄₀}-harmonic conjugate of X₁₉.