INMO 2001 Problem 1

Let $A B C$ be a triangle in which no angle is $90^{\circ}$. For any point $P$ in the plane of the triangle, let $A_{1}, B_{1}, C_{1}$ denote the reflections of $P$ in the sides $B C, C A, A B$ respectively. Prove the following statements:

(a) If $P$ is the incentre or an excentre of $A B C$, then $P$ is the circumcentre of $A_{1} B_{1} C_{1}$;

(b) If $P$ is the circumcentre of $A B C$, then $P$ is the orthocentre of $A_{1} B_{1} C_{1}$;

(c) If $P$ is the orthocentre of $A B C$, then $P$ is either the incentre or an excentre of $A_{1} B_{1} C_{1}$.