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Cyclocevian Triangle of a Point

Let P be a point and let A1B1C1 be the cevian triangle of P. The circumcircle of A1B1C1 meets each sideline in two (not necessarily distinct) points A1, A2, B1, B2 and C1, C2. Then A2B2C2 is the cyclocevian triangle of P with respect to ABC.

A1B1C1 - cevian triangle of P.
A2B2C2 - cyclocevian triangle of P.