KURSCHAK 2000 Problem 2

Let $ABC$ be a non-equilateral triangle in the plane, and let $T$ be a point different from its vertices. Define $A_T$ , $B_T$ and $C_T$ as the points where lines $AT$ , $BT$ , and $CT$ meet the circumcircle of $ABC$ . Prove that there are exactly two points $P$ and $Q$ in the plane for which the triangles $A_PB_PC_P$ and $A_QB_QC_Q$ are equilateral. Prove furthermore that line $PQ$ contains the circumcenter of $\triangle ABC$ .