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K921

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X(1)

other points below

Geometric properties :

Each cubic of the Euler pencil has an osculating circle (Ci) at X(1) whose center lies on K921.

K921 is an acnodal cubic with singularity at X(1).

It meets the circumcircle at the vertices A', B', C' of the circumcevian triangle of X(1) – aka 2nd circumperp triangle – and the points Q1, Q2, Q3 other than A, B, C of pK(X6, X145).

It has three real asymptotes parallel to the sidelines of the excentral triangle or perpendicular to the internal bisectors of ABC. These asymptotes bound a triangle A1B1C1 homothetic at X(7290) to A'B'C' and meet the cubic again at U, V, W on the satellite line (L) of the line at infinity. (L) is the trilinear polar of X(27).

The blue circle is the osculating circle at X(1) to the McCay cubic K003. Its center Ω lies on the lines {1, 513}, {100, 109}, {145, 522}, {521, 2136}, etc, now X(14812) in ETC (2017-10-10).