Circle and Lucas Cubic

This is a generalization of proposition 2 in my paper  A Note on Reflections.

On June, 10, 2014, I posed a problem at ADGEOM, which was generalized by Angel Montesdeoca in the following form:

Let ABC be a triangle. Let P be an arbitrary point on the plane of ABC. Reflect P around the vertices of cevian triangle X_aX_bX_c of a point X. This give us the triangle X'_aX'_bX'_c. Reflect the triangle

X'_aX'_bX'_c around sides of triangle ABC. Let [X_P] be the circumcircle of the triangle so formed.

P lies on the circle [X_P] if and only if X lies on the Lucas cubic.

Then, X_aX_bX_c is the pedal triangle of a point Y (on Darboux cubic) and Y is the center of [X_P].

Proof (in Spanish)