X₃₈

CROSSPOINT OF X₁ AND X₇₅

X₃₈ is the crosspoint of X₁ and X₇₅.

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

X₃₈ (CROSSPOINT OF X₁ AND X₇₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 21}, {2, 244}

• Related Centers: X₁ (INCENTER), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER), X₂₁ (SCHIFFLER POINT), X₂ (CENTROID), X₂₄₄ (-LINE CONJUGATE OF X₁₀₀)

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

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

Search Value (6, 9, 13): \(2.431598760008\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\csc A \sin(A + \omega ) : \csc B \sin(B + \omega ) : \csc C \sin(C + \omega )\)

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

• Barycentrics: \(\sin(A + \omega ) : \sin(B + \omega ) : \sin(C + \omega )\)

▸ Linear Combinations1 combination

• X₃₈ = \((3r^{2} + 8rR - s^{2})\) X₁ - \(6rR\) X₂ - \(4r^{2}\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines19 lines

{1, 21}, {2, 244}, {3, 976}, {8, 986}, {9, 614}, {10, 596}, {37, 354}, {42, 518}, {56, 201}, {57, 612}, {75, 310}, {78, 988}, {92, 240}, {99, 745}, {210, 899}, {321, 726}, {869, 980}, {912, 1064}, {1038, 1106}

▸ Functional Relations12 relations

• X₃₈ = isogonal conjugate of X₈₂

• X₃₈ = isotomic conjugate of X₃₁₁₂

• X₃₈ = anticomplement of X₁₂₁₅

• X₃₈ = crosspoint of X₁ and X₇₅

• X₃₈ = crosssum of X₁ and X₃₁

• X₃₈ = crossdifference of every pair of points on line X₆₆₁X₈₃₀

• X₃₈ = X₆₄₃-beth conjugate of X₃₈

• X₃₈ = bicentric sum of PU(35)

• X₃₈ = PU(35)-harmonic conjugate of X₆₆₁

• X₃₈ = trilinear pole of line X₂₀₈₄X₂₅₃₀

• X₃₈ = perspector of ABC and extraversion triangle of X₃₈ (which is also the anticevian triangle of X₃₈)

• X₃₈ = barycentric square root of X₈₀₄₁

Notes

X₃₈ is the {X₁,X₆₃}-harmonic conjugate of X₃₁.