X₃₁₆₃

X₂-CEVA CONJUGATE OF X₃₀

X₃₁₆₃ = midpoint of X₂ ane X₆₄₈

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

X₃₁₆₃ (X₂-CEVA CONJUGATE OF X₃₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 13}, {2, 648}

• Related Centers: X₂ (CENTROID), X₃₀ (EULER INFINITY POINT), X₆ (SYMMEDIAN POINT), X₁₃ (1st ISOGONIC CENTER), X₆₄₈ (TRILINEAR POLE OF EULER LINE)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₂ and \(u : v : w\) = X₃₀

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\), where \(f(a,b,c) = u(-u/x + v/y + w/z)\)

Search Value (6, 9, 13): \(0.833100333097\)

▸ Alternate Coordinates1 form

• Barycentrics: \((\sin^{2} A)(\cos A - 2 \cos B \cos C)^{2} : :\)

X₃₁₆₃ lies on the Steiner inellipse.

▸ Incident Central Lines11 lines

{2, 648}, {6, 13}, {30, 1990}, {216, 549}, {233, 547}, {376, 577}, {553, 1086}, {1084, 1196}, {1100, 1146}, {1636, 1637}, {2482, 2799}

▸ Functional Relations15 relations

• X₃₁₆₃ = midpoint of X₂ ane X₆₄₈

• X₃₁₆₃ = isotomic conjugate of X₃₁₆₂₁

• X₃₁₆₃ = complement of X₁₄₉₄

• X₃₁₆₃ = isogonal conjugate of isotomic conjugate of X₃₆₇₈₉

• X₃₁₆₃ = X_i-Ceva conjugate of X_j for these (i,j): (2, 30), (30, 3081)

• X₃₁₆₃ = crosspoint of X₃₆₂₉₈ and X₃₆₂₉₉

• X₃₁₆₃ = X₃₀₈₁-cross conjugate of X₃₀

• X₃₁₆₃ = crosspoint of X₂ and X₃₀

• X₃₁₆₃ = crosssum of X₆ and X₇₄

• X₃₁₆₃ = crossdifference of every pair of points on the line X₇₄X₅₂₆

• X₃₁₆₃ = center of circumconic that is locus of trilinear poles of lines parallel to Euler line (i.e. lines that pass through X₃₀)

• X₃₁₆₃ = perspector of circumconic centered at X₃₀ (parabola {A, B, C,X₃₀,X₄₇₆})

• X₃₁₆₃ = perspector of ABC and medial triangle of cevian triangle of X₃₀

• X₃₁₆₃ = barycentric square of X₃₀

• X₃₁₆₃ = X₁₂₀₃₄-of-orthic-triangle if ABC is acute