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 that
(i) If P is the incenter or an excentre of ABC, then P is the circumenter of A1B1C1;
(ii) If P is the circumcentre of ABC, then P is the orthocentre of A1B1C1;
(iii) If P is the orthocentre of ABC, then P is either the incentre or an excentre of A1B1C1. geometrygeometric transformationreflectionincenterparallelogramcircumcircleratio