X₆₉₆

EVEN (- 4, - 3) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₆₉₆ lies on the line at infinity. The first trilinear coordinate has the form

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

X₆₉₆ (EVEN (- 4, - 3) INFINITY POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {30, 511}, {313, 561}

• Related Centers: X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈), X₃₁₃ (ISOTOMIC CONJUGATE OF X₅₈), X₅₆₁ (ISOGONAL CONJUGATE OF 4th POWER POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a ^{-5}(b ^{-3} + c ^{-3}) - a ^{-4}(b ^{-4} + c ^{-4})\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)

▸ Incident Central Lines2 lines

{30, 511}, {313, 561}

▸ Functional Relations13 relations

• X₅₃₈ = even (- 2, 0) infinity point

• X₅₃₆ = even (- 1, 0) infinity point

• X₅₁₉ = even (0, 1) infinity point

• X₁₀₆ = even (0, 1) circumcircle point

• X₅₂₄ = even (0, 2) infinity point

• X₁₁₁ = even (0, 2) circumcircle point

• X₅₁₈ = even (1, 2) infinity point

• X₁₀₅ = even (1, 2) circumcircle point

• X₆₇₄ = even (2, 3) infinity point

• X₆₇₅ = even (2, 3) circumcircle point

• X₅₁₁ = even (2, 4) infinity point

• X₉₈ = even (2, 4) circumcircle point

• X₆₉₆ = isogonal conjugate of X₆₉₇

Notes

\(a^{m-1}(b^{n} + c^{n})\) - \(a^{n-1}(b^{m} + c^{m})\).

If m and n are distinct integers, this form fits the definition of even polynomial center as in Clark Kimberling, “Functional equations associated with triangle geometry,” Aequationes Mathematicae 45 (1993) 127-152. This form, perhaps appearing initially here (July 7, 2001) defines a triangle center for arbitrary distinct real numbers m and n. Selected even infinity and circumcircle points begin at X₆₉₆; odd ones begin at X₇₆₈.

Certain points of this type occur prior to this section. They are as follows: