X₃₁₁₀

ORTHOGONAL PROJECTION OF X₁ ON BROCARD AXIS

X₃₁₁₀ = crossdifference of every pair of points on the line X₅₂₃X₂₂₃₈

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

X₃₁₁₀ (ORTHOGONAL PROJECTION OF X₁ ON BROCARD AXIS) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Central Lines: {3, 3286}

• Related Centers: X₃ (CIRCUMCENTER), X₃₂₈₆ (INTERSECTION OF LINES X)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(b^{4} + c^{4} + 2a^{2}bc - a(b + c)(b^{2} + c^{2}))/(b + c)\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(1.581407675657\)

X₃₁₁₀ lies on the circle O(1,3).

▸ Incident Central Lines8 lines

{1, 512}, {3, 6}, {21, 1083}, {60, 249}, {691, 840}, {741, 1015}, {813, 1500}, {1362, 1414}

▸ Functional Relations3 relations

• X₃₁₁₀ = inverse-in-circumcircle of X₃₂₈₆

• X₃₁₁₀ = crossdifference of every pair of points on the line X₅₂₃X₂₂₃₈

• X₃₁₁₀ = X₉₉ of X₁-Brocard triangle