Let ABC be a triangle in which no angle is 90◦. For any point P in the plane
of the triangle, let A1,B1,C1 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 A1B1C1;
(b) If P is the circumcentre of ABC, then P is the orthocentre of A1B1C1;
(c) If P is the orthocentre of ABC, then P is either the incentre or an excentre of
A1B1C1.