X₇₃

CROSSPOINT OF INCENTER AND CIRCUMCENTER

X₇₃ is the crosspoint of incenter and circumcenter.

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

X₇₃ (CROSSPOINT OF INCENTER AND CIRCUMCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 4}, {6, 41}

• Related Centers: X₁ (INCENTER), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₆ (SYMMEDIAN POINT), X₄₁ (-CEVA CONJUGATE OF X₃₁)

Trilinears: \(\sec B + \sec C : \sec C + \sec A : \sec A + \sec B\)

Barycentrics: \((\cos B + \cos C) \sin 2A : (\cos C + \cos A) \sin 2B : (\cos A + \cos B) \sin 2C\)

Search Value (6, 9, 13): \(-2.306123931414\)

▸ Alternate Coordinates1 form

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

▸ Linear Combinations1 combination

• X₇₃ = \((r^{2} - 4R^{2} + s^{2})\) X₁ - \(6rR\) X₂ + \(4rR\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines20 lines

{1, 4}, {3, 212}, {6, 41}, {21, 651}, {35, 74}, {36, 54}, {37, 836}, {42, 65}, {55, 64}, {57, 386}, {66, 976}, {68, 1060}, {69, 77}, {72, 201}, {102, 947}, {228, 408}, {284, 951}, {290, 336}, {1036, 1037}, {1057, 1059}

▸ Functional Relations16 relations

• X₇₃ = isogonal conjugate of X₂₉

• X₇₃ = anticomplement of X₃₄₃₈₁

• X₇₃ = X₁-Ceva conjugate of X₆₅

• X₇₃ = X₂₂₈-cross conjugate of X₇₁

• X₇₃ = crosspoint of X_i and X_j for these (i,j): (1, 3), (77, 222), (226, 307)

• X₇₃ = crosssum of X_i and X_j for these (i,j): (1, 4), (33, 281)

• X₇₃ = crossdifference of every pair of points on line X₂₄₃X₅₂₂

• X₇₃ = X_i-Hirst inverse of X_j for these (i,j): (1, 243), (65, 851)

• X₇₃ = X_i-beth conjugate of X_j for these (i,j): (1, 1042), (3, 73), (21, 946), (72, 72), (100, 10), (101, 73), (295, 73)

• X₇₃ = bicentric sum of PU(16)

• X₇₃ = PU(16)-harmonic conjugate of X₆₅₂

• X₇₃ = trilinear product of PU(83)

• X₇₃ = trilinear pole of line X₆₄₇X₈₂₂

• X₇₃ = X₆₃-isoconjugate of X₈₇₄₈

• X₇₃ = X₉₂-isoconjugate of X₂₈₄

• X₇₃ = {X₁,X₁₇₄₅}-harmonic conjugate of X₄

Notes

X₇₃ is the {X₁₀₆₄,X₁₀₆₆}-harmonic conjugate of X₁.