X₃₄₂₁

ISOGONAL CONJUGATE OF X₃₄₂₀

X₃₄₂₁ is the isogonal conjugate of X₃₄₂₀.

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

X₃₄₂₁ (ISOGONAL CONJUGATE OF X₃₄₂₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {8, 329}

• Related Centers: X₈ (NAGEL POINT), X₃₂₉ (ISOTOMIC CONJUGATE OF X₁₈₉)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = 1/x(3420)\)

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

Search Value (6, 9, 13): \(5.817342576491\)

▸ Alternate Coordinates1 form

• Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{4} - 4abc(b + c - a) - (b^{2} - c^{2})^{2} (E\). Danneels)

▸ Incident Central Lines15 lines

{1, 2551}, {2, 495}, {4, 8}, {40, 2123}, {57, 388}, {100, 376}, {144, 153}, {145, 1058}, {150, 668}, {200, 515}, {219, 1249}, {497, 519}, {527, 1478}, {529, 1376}, {631, 2975}

▸ Functional Relations4 relations

• X₃₄₂₁ = midpoint of X₈ and X₃₂₉

• X₃₄₂₁ = reflection of X_i in X_j for these (i,j): (57, 10), (2096, 3359)

• X₃₄₂₁ = isogonal conjugate of X₃₄₂₀

• X₃₄₂₁ = anticomplement of X₉₉₉