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)

Concurrent Circles

Let ABC be a triangle.
Let P be a point on the plane of ABC.
Let a, b, c, are the sides opposite vertices A, B, C.
Let P_a be the reflection of P around the perpendicular
bisector of a. Define P_b, P_c cyclically.

Then, circles (AP_bP_c), (BP_aP_c), (CP_aP_b) are concurrent at
the circumcircle of ABC.

Proof