X₃₅₆₃

1st MOSES CIRCUMCIRCLE POINT

X₃₅₆₃ = reflection of X₃₅₆₅ in X₃

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

X₃₅₆₃ (1st MOSES CIRCUMCIRCLE POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3565, 3}

• Related Centers: X₃₅₆₅ (2nd MOSES CIRCUMCIRCLE POINT), X₃ (CIRCUMCENTER)

Trilinears: \((\sec A)/(b^{2} \cos^{2} C + c^{2} \cos^{2} B - bc \cos A) : :\)

Search Value (6, 9, 13): \(11.115434509251\)

▸ Alternate Coordinates1 form

• Trilinears: \(a/[(b^{2} + c^{2} - a^{2})(2a^{4} + b^{4} + c^{4} - 2b^{2}c^{2} - c^{2} a^{2} - a^{2}b^{2})] : :\)

X₃₅₆₃ lies on the circumcircle.

▸ Incident Central Lines15 lines

{2, 136}, {3, 2971}, {4, 99}, {24, 112}, {25, 110}, {100, 1824}, {101, 2333}, {107, 459}, {186, 691}, {232, 1692}, {378, 1296}, {403, 935}, {427, 930}, {468, 476}, {934, 1426}

▸ Functional Relations15 relations

• X₃₅₆₃ = reflection of X₃₅₆₅ in X₃

• X₃₅₆₃ = isogonal conjugate of X₃₅₆₄

• X₃₅₆₃ = anticomplement of X₃₁₈₄₂

• X₃₅₆₃ = cevapoint of X₂₅ and X₂₃₂

• X₃₅₆₃ = Λ(X₅, X₆)

• X₃₅₆₃ = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A, B, C,X₂,X₂₄}

• X₃₅₆₃ = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A, B, C,X₄,X₂₅}

• X₃₅₆₃ = inverse-in-polar-circle of X₁₁₄

• X₃₅₆₃ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₃₆

• X₃₅₆₃ = trilinear pole of line X₆X₉₂₄

• X₃₅₆₃ = trilinear pole, wrt Thomson triangle, of line X₁₈₂X₃₁₆₇

• X₃₅₆₃ = Ψ(X₆, X₉₂₄)

• X₃₅₆₃ = Cundy-Parry Phi transform of X₁₄₂₄₈

• X₃₅₆₃ = Cundy-Parry Psi transform of X₆₃₃₇

• X₃₅₆₃ = perspector, wrt dual of orthic triangle, of polar circle

Notes

The antipodal pair X₃₅₆₃ to X₃₅₆₅ were described by Peter Moses, Dec. 13, 2004.

Let A' be the reflection in line BC of the A-vertex of the antipedal triangle of X₆, and define B' and C' cyclically. Let O_A be the circumcenter of AB'C', and define O_B and O_C cyclically. Let U be the circumcenter of A'BC, and define V and W cyclically. Let O' be the circumcenter of O_AO_BO_C. Let O″ be the circumcenter of UVW. Then X₃₅₆₃ = Λ(O',O″). Also, O'O″∩(infinity line) = X₃₅₆₄. (Randy Hutson, June 19, 2015)