Question 69
Let $ABC$ 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 $BC, CA, AB$ respectively. Prove the following statements:
a. If $P$ is the incentre or an excentre of $ABC$, then $P$ is the circumcentre of $A_{1}B_{1}C_{1}$;
b. If $P$ is the circumcentre of $ABC$, then $P$ is the orthocentre of $A_{1}B_{1}C_{1}$;
c. If $P$ is the orthocentre of $ABC$, then $P$ is either the incentre or an excentre of $A_{1}B_{1}C_{1}$.
a. If $P$ is the incentre or an excentre of $ABC$, then $P$ is the circumcentre of $A_{1}B_{1}C_{1}$;
b. If $P$ is the circumcentre of $ABC$, then $P$ is the orthocentre of $A_{1}B_{1}C_{1}$;
c. If $P$ is the orthocentre of $ABC$, then $P$ is either the incentre or an excentre of $A_{1}B_{1}C_{1}$.