X₃₂₂₈

SS(a → bc) OF X₁₁₁

X₃₂₂₈ = X₁₆₄₅-cross conjugate of X₅₁₂

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

X₃₂₂₈ (SS(a → bc) OF X₁₁₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 99}, {42, 190}

• Related Centers: X₃₂₃₁ (ISOGONAL CONJUGATE OF X₃₂₂₈), X₆ (SYMMEDIAN POINT), X₉₉ (STEINER POINT), X₄₂ (CROSSPOINT OF INCENTER AND SYMMEDIAN POINT), X₁₉₀ (YFF PARABOLIC POINT)

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

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

Search Value (6, 9, 13): \(-1.768531538174\)

▸ Alternate Coordinates1 form

• Trilinears: for X₃₂₂₈ are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X₁₁₁. As a symbolic substitution, this mapping takes circumconics to circumconics. In particular, it maps circumcircle onto the Steiner circumellipse, which passes through X₃₂₂₈.

X₃₂₂₈ lies on the Steiner circumellipse.

▸ Incident Central Lines9 lines

{6, 99}, {25, 648}, {37, 668}, {42, 190}, {111, 385}, {263, 1992}, {290, 2395}, {538, 886}, {671, 804}

▸ Functional Relations13 relations

• X₃₂₂₈ = reflection of X_i in X_j for these (i,j): (2, 1084), (670, 2)

• X₃₂₂₈ = isogonal conjugate of X₃₂₃₁

• X₃₂₂₈ = isotomic conjugate of X₅₃₈

• X₃₂₂₈ = complement of X₃₉₃₆₁

• X₃₂₂₈ = anticomplement of X₃₅₀₇₃

• X₃₂₂₈ = X₁₆₄₅-cross conjugate of X₅₁₂

• X₃₂₂₈ = crossdifference of every pair of points on the line X₈₈₇X₈₈₈

• X₃₂₂₈ = Steiner-circumellipse-antipode of X₆₇₀

• X₃₂₂₈ = Steiner-circumellipse-X₆-antipode of X₉₉

• X₃₂₂₈ = projection from Steiner inellipse to Steiner circumellipse of X₁₀₈₄

• X₃₂₂₈ = the point of intersection, other than A, B, and C, of the Steiner circumellipse and hyperbola {{A, B, C,X₂,X₆}}

• X₃₂₂₈ = antipode of X₂ in hyperbola {{A, B, C,X₂,X₆}}

• X₃₂₂₈ = trilinear pole of line X₂X₅₁₂