X₇₆₉–X₈₄₃

Odd o-(p, q) Circumcircle Points

Odd trigonometric o-(p, q) points on the circumcircle and line at infinity, paired with their isogonal conjugates.

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

X₇₆₉–X₈₄₃ (Odd o-(p, q) Circumcircle Points) — 16 representative centers (from 75 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X₇₆₉o-(- 4, - 3) CIRCUMCIRCLE POINT

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

hence the name “(m, n)-circumcircle point”.

X₇₇₀POINT ACAMAR

X₇₇₀ = crosssum of X₁ and X₇₇₀

X₇₇₁ISOGONAL CONJUGATE OF X₇₇₀

X₇₇₁ is the isogonal conjugate of X₇₇₀.

X₇₇₂ODD (- 4, - 1) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₇₂ lies on the line at infinity.

X₇₇₃o-(- 4, - 1) CIRCUMCIRCLE POINT

X₇₇₃ = isogonal conjugate of X₇₇₂

X₇₇₄CROSSPOINT OF X₁ AND X₁₅₈

X₇₇₄ is the crosspoint of X₁ and X₁₅₈.

X₇₇₅ISOGONAL CONJUGATE OF X₇₇₄

X₇₇₅ is the isogonal conjugate of X₇₇₄.

X₇₇₆ODD (- 4, 1) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₇₆ lies on the line at infinity.

X₇₇₇o-(- 4, 1) CIRCUMCIRCLE POINT

X₇₇₇ = isogonal conjugate of X₇₇₆

X₇₇₈ODD (- 4, 2) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₇₈ lies on the line at infinity.

X₇₇₉o-(- 4, 2) CIRCUMCIRCLE POINT

X₇₇₉ = isogonal conjugate of X₇₇₈

X₇₈₀ODD (- 4, 3) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₈₀ lies on the line at infinity.

X₇₈₁o-(- 4, 3) CIRCUMCIRCLE POINT

X₇₈₁ = isogonal conjugate of X₇₈₀

X₇₈₂ODD (- 4, 4) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₈₂ lies on the line at infinity.

X₇₈₃o-(- 4, 4) CIRCUMCIRCLE POINT

X₇₈₃ = isogonal conjugate of X₇₈₂

X₇₈₄ODD (- 3, - 2) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₈₄ lies on the line at infinity.

X₇₈₅o-(- 3, - 2) CIRCUMCIRCLE POINT

X₇₈₅ = Ψ(X₇₆, X₃₇)

X₇₈₆ODD (- 3, - 1) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₈₆ lies on the line at infinity.

X₇₈₇o-(- 3, - 1) CIRCUMCIRCLE POINT

X₇₈₇ = isogonal conjugate of X₇₈₆

X₇₈₈ODD (- 3, 0) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₈₈ lies on the line at infinity.

X₇₈₉o-(- 3, 0) CIRCUMCIRCLE POINT

X₇₈₉ = intersection, other than A, B, C, of circumcircle and hyperbola {A, B, C,PU(6)}}

X₇₉₀ODD (- 3, 1) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₉₀ lies on the line at infinity.

X₇₉₁o-(- 3, 1) CIRCUMCIRCLE POINT

X₇₉₁ = isogonal conjugate of X₇₉₀

X₇₉₂ODD (- 3, 2) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₉₂ lies on the line at infinity.

X₇₉₃o-(- 3, 2) CIRCUMCIRCLE POINT

X₇₉₃ = isogonal conjugate of X₇₉₂

X₇₉₄ODD (- 3, 3) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₉₄ lies on the line at infinity.

X₇₉₅o-(- 3, 3) CIRCUMCIRCLE POINT

X₇₉₅ = isogonal conjugate of X₇₉₄

X₇₉₆ODD (- 3, 4) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₇₉₆ lies on the line at infinity.

X₇₉₇o-(- 3, 4) CIRCUMCIRCLE POINT

X₇₉₇ = isogonal conjugate of X₇₉₆

X₇₉₈CROSSDIFFERENCE OF X₁ AND X₇₅

X₇₉₈ = X₁₆₃-Ceva conjugate of X₃₁

X₇₉₉ISOGONAL CONJUGATE OF X₇₉₈

X₇₉₉ is the isogonal conjugate of X₇₉₈.

X₈₀₀CROSSPOINT OF X₂ AND X₆₄

X₈₀₀ is the crosspoint of X₂ and X₆₄.

X₈₀₁ISOGONAL CONJUGATE OF X₈₀₀

X₈₀₁ is the isogonal conjugate of X₈₀₀.

X₈₀₂ODD (- 2, 1) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₀₂ lies on the line at infinity.

X₈₀₃o-(- 2, 1) CIRCUMCIRCLE POINT

X₈₀₃ = isogonal conjugate of X₈₀₂

X₈₀₄ODD (- 2, 2) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₀₄ lies on the line at infinity.

X₈₀₅o-(- 2, 2) CIRCUMCIRCLE POINT

X₈₀₅ = trilinear pole of line X₆X₆₉₄

X₈₀₆ODD (- 2, 3) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₀₆ lies on the line at infinity.

X₈₀₇o-(- 2, 3) CIRCUMCIRCLE POINT

X₈₀₇ = isogonal conjugate of X₈₀₆

X₈₀₈ODD (- 2, 4) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₀₈ lies on the line at infinity.

X₈₀₉o-(- 2, 4) CIRCUMCIRCLE POINT

X₈₀₉ = isogonal conjugate of X₈₀₈

X₈₁₀CROSSPOINT OF X₁ AND X₁₆₃

X₈₁₀ is the crosspoint of X₁ and X₁₆₃.

X₈₁₁ISOGONAL CONJUGATE OF X₈₁₀

X₈₁₁ is the isogonal conjugate of X₈₁₀.

X₈₁₂ODD (- 1, 1) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₁₂ lies on the line at infinity.

X₈₁₃o-(- 1, 1) CIRCUMCIRCLE POINT

X₈₁₃ = trilinear pole of line X₆X₂₉₂

X₈₁₄ODD (- 1, 2) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₁₄ lies on the line at infinity.

X₈₁₅o-(- 1, 2) CIRCUMCIRCLE POINT

X₈₁₅ = isogonal conjugate of X₈₁₄

X₈₁₆ODD (- 1, 3) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₁₆ lies on the line at infinity.

X₈₁₇o-(- 1, 3) CIRCUMCIRCLE POINT

X₈₁₇ = isogonal conjugate of X₈₁₆

X₈₁₈ODD (- 1, 4) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₁₈ lies on the line at infinity.

X₈₁₉o-(- 1, 4) CIRCUMCIRCLE POINT

X₈₁₉ = isogonal conjugate of X₈₁₈

X₈₂₀CROSSPOINT OF X₁ AND X₂₅₅

X₈₂₀ is the crosspoint of X₁ and X₂₅₅.

X₈₂₁ISOGONAL CONJUGATE OF X₈₂₀

X₈₂₁ is the isogonal conjugate of X₈₂₀.

X₈₂₂CROSSDIFFERENCE OF X₁ AND X₂₉

X₈₂₂ = X₁₆₃-Ceva conjugate of X₄₈

X₈₂₃ISOGONAL CONJUGATE OF X₈₂₂

X₈₂₃ is the isogonal conjugate of X₈₂₂.

X₈₂₄ODD (0, 3) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₂₄ lies on the line at infinity.

X₈₂₅o-(0, 3) CIRCUMCIRCLE POINT

X₈₂₅ = intersection, other than A, B, C, of circumcircle and hyperbola {A, B, C,PU(12)}}

X₈₂₆ODD (0, 4) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₂₆ lies on the line at infinity.

X₈₂₇o-(0, 4) CIRCUMCIRCLE POINT

X₈₂₇ = trilinear pole of line X₆X₂₂

X₈₂₈CROSSPOINT OF X₂ AND X₂₅₅

X₈₂₈ is the crosspoint of X₂ and X₂₅₅.

X₈₂₉ISOGONAL CONJUGATE OF X₈₂₈

X₈₂₉ is the isogonal conjugate of X₈₂₈.

X₈₃₀ODD (1, 3) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₃₀ lies on the line at infinity.

X₈₃₁o-(1, 3) CIRCUMCIRCLE POINT

X₈₃₁ = trilinear pole of line X₆X₃₈

X₈₃₂ODD (1, 4) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₃₂ lies on the line at infinity.

X₈₃₃o-(1, 4) CIRCUMCIRCLE POINT

X₈₃₃ = isogonal conjugate of X₈₃₂

X₈₃₄ODD (2, 3) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₃₄ lies on the line at infinity.

X₈₃₅o-(2, 3) CIRCUMCIRCLE POINT

X₈₃₅ = trilinear pole of line X₆X₁₀

X₈₃₆CROSSPOINT OF X₁ AND X₃₉₄

X₈₃₆ is the crosspoint of X₁ and X₃₉₄.

X₈₃₇ISOGONAL CONJUGATE OF X₈₃₆

X₈₃₇ is the isogonal conjugate of X₈₃₆.

X₈₃₈ODD (3, 4) INFINITY POINT

As the isogonal conjugate of a point on the circumcircle, X₈₃₈ lies on the line at infinity.

X₈₃₉o-(3, 4) CIRCUMCIRCLE POINT

X₈₃₉ = trilinear pole of line X₆X₃₂₁

X₈₄₀ISOGONAL CONJUGATE OF X₅₂₈

X₈₄₀ is the isogonal conjugate of X₅₂₈.

X₈₄₁ISOGONAL CONJUGATE OF X₅₄₁

X₈₄₁ is the isogonal conjugate of X₅₄₁.

X₈₄₂ISOGONAL CONJUGATE OF X₅₄₂

X₈₄₂ is the antipode of X₆₉₁ on the circumcircle.

X₈₄₃ISOGONAL CONJUGATE OF X₅₄₃

X₈₄₃ is the isogonal conjugate of X₅₄₃.