X₃₂₂₆

SS(a → bc) OF X₁₀₅

X₃₂₂₆ = cevapoint of X₁ and X₃₂₉

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

X₃₂₂₆ (SS(a → bc) OF X₁₀₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {58, 99}, {75, 87}

• Related Centers: X₃₀₀₉ ((LEMOINE AXIS)∩(NAGEL LINE)), X₅₈ (ISOGONAL CONJUGATE OF X₁₀), X₉₉ (STEINER POINT), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER), X₈₇ (-CROSS CONJUGATE OF X₁)

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

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

Search Value (6, 9, 13): \(-2.273736534287\)

▸ Alternate Coordinates1 form

• Trilinears: for X₃₂₂₆ are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X₁₀₅. As a symbolic substitution, this mapping takes circumconics to circumconics. In particular, it maps circumcircle onto the Steiner circumellipse, which passes through X₃₂₂₆.

X₃₂₂₆ lies on the Steiner circumellipse.

▸ Incident Central Lines10 lines

{1, 668}, {6, 190}, {56, 664}, {58, 99}, {75, 87}, {86, 670}, {239, 292}, {648, 1474}, {666, 1438}, {1027, 2481}

▸ Functional Relations9 relations

• X₃₂₂₆ = isogonal conjugate of X₃₀₀₉

• X₃₂₂₆ = isotomic conjugate of X₇₂₆

• X₃₂₂₆ = complement of X₃₉₃₅₄

• X₃₂₂₆ = anticomplement of X₂₀₅₃₂

• X₃₂₂₆ = cevapoint of X₁ and X₃₂₉

• X₃₂₂₆ = X_i-cross conjugate of X_j for these (i,j): (291, 673), (350, 86), (659, 190)

• X₃₂₂₆ = trilinear pole of line X₂X₆₄₉

• X₃₂₂₆ = Steiner-circumellipse-X₁-antipode of X₆₆₈

• X₃₂₂₆ = Steiner-circumellipse-X₆-antipode of X₁₉₀