X₃₂₂₁

SS(a → bc) OF X₆₄₇

As the isogonal conjugate of a point on the circumcircle, X₃₂₂₁ lies on the line at infinity.

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

X₃₂₂₁ (SS(a → bc) OF X₆₄₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {30, 511}, {882, 1843}

• Related Centers: X₃₂₂₂ (ISOGONAL CONJUGATE OF X₃₂₂₁), X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈), X₈₈₂ (ISOTOMIC CONJUGATE OF X₈₈₀), X₁₈₄₃ (ZOSMA TRANSFORM OF X₇₅)

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

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

▸ Alternate Coordinates1 form

• Trilinears: for X₃₂₂₁ are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X₆₄₇. As a symbolic substitution, this mapping takes lines onto lines. In particular, points on the Euler line, X₂X₃, with coefficients given by X₆₄₇, are mapped to the line X₆X₁₉₄, with coefficients given by X₃₂₂₁. Symbolic substitutions are introduced in the following article:

▸ Incident Central Lines3 lines

{30, 511}, {669, 2451}, {882, 1843}

▸ Functional Relations4 relations

• X₃₂₂₁ = isogonal conjugate of X₃₂₂₂

• X₃₂₂₁ = perspector of hyperbola {{A, B, C,X₂,X₁₉₄}}

• X₃₂₂₁ = bicentric difference of PU(154)

• X₃₂₂₁ = ideal point of PU(154)

Notes

C. Kimberling, “Symbolic substitutions in the transfigured plane of a triangle,” Aequationes Mathematicae 73 (2007) 156-171.