X₃₄₉₆

INTERSECTION OF LINES X₁X_32 AND X₄X_9

X₃₄₉₆ = crossdifference of every pair of points on the line X₁₄₅₉X₁₄₉₁

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

X₃₄₉₆ (INTERSECTION OF LINES X₁X_32 AND X₄X_9) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 9}, {1, 32}

• Related Centers: X₃₄₉₇ (ISOGONAL CONJUGATE OF X₃₄₉₆), X₄ (ORTHOCENTER), X₉ (MITTENPUNKT), X₁ (INCENTER), X₃₂ (3rd POWER POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = b^{3} + c^{3} - a^{3} + abc (E\). Danneels)

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

Search Value (6, 9, 13): \(4.640916017918\)

X₃₄₉₆ lies on the cubic Z(X₃₈₄).

▸ Incident Central Lines13 lines

{1, 32}, {4, 9}, {6, 986}, {39, 893}, {57, 348}, {63, 194}, {191, 2795}, {220, 2943}, {257, 384}, {517, 2329}, {910, 960}, {2170, 2975}, {2288, 2323}

▸ Functional Relations3 relations

• X₃₄₉₆ = isogonal conjugate of X₃₄₉₇

• X₃₄₉₆ = complement of X₃₃₈₆₇

• X₃₄₉₆ = crossdifference of every pair of points on the line X₁₄₅₉X₁₄₉₁