A Theorem about Simultaneous Orthological and Homological Triangles

In this paper we prove that if $P_1, P_2$ are isogonal points in the triangle $ABC$, and if $A_1B_1C_1$ and $A_2B_2C_2$ are their corresponding pedal triangles such that the triangles $ABC$ and $A_1B_1C_1$ are homological (the lines $AA_1, BB_1, CC_1$ are concurrent), then the triangles $ABC$ and $A_2B_2C_2$ are also homological.