X₃₂₂₇

SS(a → bc) OF X₁₀₆

X₃₂₂₇ = Steiner-circumellipse-X₁-antipode of X₁₉₀

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

X₃₂₂₇ (SS(a → bc) OF X₁₀₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {81, 99}, {1, 190}

• Related Centers: X₃₂₃₀ (ISOGONAL CONJUGATE OF X₃₂₂₇), X₈₁ (CEVAPOINT OF INCENTER AND SYMMEDIAN POINT), X₉₉ (STEINER POINT), X₁ (INCENTER), X₁₉₀ (YFF PARABOLIC POINT)

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

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

Search Value (6, 9, 13): \(-1.131330786086\)

▸ Alternate Coordinates1 form

• Trilinears: for X₃₂₂₇ are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X₁₀₆. As a symbolic substitution, this mapping takes circumconics to circumconics. In particular, it maps circumcircle onto the Steiner circumellipse, which passes through X₃₂₂₇.

X₃₂₂₇ lies on the Steiner circumellipse.

▸ Incident Central Lines13 lines

{1, 190}, {2, 668}, {28, 648}, {57, 664}, {81, 99}, {88, 239}, {105, 666}, {274, 670}, {291, 519}, {335, 2087}, {671, 2787}, {903, 1022}, {957, 1992}

▸ Functional Relations12 relations

• X₃₂₂₇ = reflection of X_i in X_j for these (i,j): (2, 1015), (668, 2)

• X₃₂₂₇ = isogonal conjugate of X₃₂₃₀

• X₃₂₂₇ = isotomic conjugate of X₅₃₆

• X₃₂₂₇ = complement of X₃₉₃₆₀

• X₃₂₂₇ = anticomplement of X₁₃₄₆₆

• X₃₂₂₇ = Steiner-circumellipse-X₁-antipode of X₁₉₀

• X₃₂₂₇ = X₁₆₄₆-cross conjugate of X₅₁₃

• X₃₂₂₇ = Steiner-circumellipse-antipode of X₆₆₈

• X₃₂₂₇ = projection from Steiner inellipse to Steiner circumellipse of X₁₀₁₅

• X₃₂₂₇ = the point of intersection, other than A, B, and C, of the Steiner circumellipse and hyperbola {{A, B, C,X₁,X₂}}

• X₃₂₂₇ = antipode of X₂ in hyperbola {{A, B, C,X₁,X₂}}

• X₃₂₂₇ = trilinear pole of line X₂X₅₁₃