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UVW : cevian triangle of P694 = X(14565)

A', B', C' tangentials of A, B, C

Q1, Q2, Q3 on the circumcircle

Q'1, Q'2, Q'3 their tangentials

K694 is another example of cubic meeting the circumcircle (O) at six points with concurrent tangent, here at X(512). See the related K444, K579, K693.

In this case, the 6 tangents are parallel and perpendicular to the Brocard axis. The pseudo-pivot P694 of K694 is the isogonal conjugate the homothetic of X(115) under h(X6, -2) with SEARCH = 5.56486238128672, now X(14565) in ETC (2017-09-28).

K694 meets the circumcircle at Q1, Q2, Q3 which are the vertices of an equilateral triangle. This triangle is not related to the usual equilateral triangles.

The satellite conic (S) of (O) is bitangent to (O) at X(1379), X(1380) and also bitangent to C(O, 2R) at its intersections M1, M2 with the perpendicular at O to the Brocard axis.

See Cubics Related to Coaxial Circles ยง10.4 and Circumcubics meeting the Circumcircle at six points with concurring tangents.

See also Pseudo-Pivotal Cubics and Poristic Triangles.