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X(3), X(4)

midpoints of ABC

Denote by A1, B1, C1 the reflections of A, B, C in P and by A2, B2, C2 the reflections of P in the sidelines BC, CA, AB.

The three circles AB1C1, BC1A1, CA1B1 have a common point D on the circumcircle.

The three circles AB2C2, BC2A2, CA2B2 have also a common point N on the circumcircle.

These points D, N coincide if and only if P lies on the Musselman (first) cubic K026 (Musselman, Some loci connected with a triangle. Monthly, p.354-361, June-July 1940).

On the other hand, they are antipodes if and only if P lies on the cubic K226 we call Musselman (fourth) cubic.

K226 is an equilateral cubic. Its three asymptotes are perpendicular to the sidelines of the Morley triangle and form an equilateral triangle with center X(5), the nine-point center, whose circumcircle has radius R/3.