X₇₆₈

ODD (- 4, - 3) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₆₈ lies on the line at infinity. The first trilinear coordinate has the form

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

X₇₆₈ (ODD (- 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}

• Related Centers: X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈)

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 Lines1 line

{30, 511}

▸ Functional Relations18 relations

• X₅₂₃ = odd (- 4, - 2) infinity point

• X₆₈₈ = odd (- 4, 0) infinity point

• X₆₈₉ = o-(- 4, 0) circumcircle point

• X₅₁₄ = odd (- 2, - 1) infinity point

• X₁₀₁ = o-(- 2, - 1) circumcircle point

• X₅₁₂ = odd (- 2, 0) infinity point

• X₉₉ = o-(- 2, 0) circumcircle point

• X₅₁₃ = odd (- 1, 0) infinity point

• X₁₀₀ = o-(- 1, 0) circumcircle point

• X₅₁₄ = odd (0, 1) infinity point

• X₁₀₁ = o-(0, 1) circumcircle point

• X₅₂₃ = odd (0, 2) infinity point

• X₁₁₀ = o-(0, 2) circumcircle point

• X₅₁₃ = odd (1, 2) infinity point

• X₁₀₀ = o-(1, 2) circumcircle point

• X₅₁₂ = odd (2, 4) infinity point

• X₉₉ = o-(2, 4) circumcircle point

• X₇₆₈ = isogonal conjugate of X₇₆₉

Notes

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

corresponding to an odd polynomial center in case m and n are distinct integers. See the note accompanying X₆₉₆, where even (m,n) infinity points and even (m,n) circumcircle points are introduced. [For nonzero n, “odd (m,n) circumcircle point” would be a misnomer (as the point is an even polynomial center); consequently, the prefix o- is used to distinguish this point from “even (m,n) circumcircle point” defined at X₆₉₆.] Certain points of these classes occur prior to this section. They are as follows: