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Triangles

Source: INMO 2001 Problem 1

October 10, 2005
geometrygeometric transformationreflectionincenterparallelogramcircumcircleratio

Problem Statement

Let ABCABC be a triangle in which no angle is 90∘90^{\circ}. For any point PP in the plane of the triangle, let A1,B1,C1A_1, B_1, C_1 denote the reflections of PP in the sides BC,CA,ABBC,CA,AB respectively. Prove that (i) If PP is the incenter or an excentre of ABCABC, then PP is the circumenter of A1B1C1A_1B_1C_1; (ii) If PP is the circumcentre of ABCABC, then PP is the orthocentre of A1B1C1A_1B_1C_1; (iii) If PP is the orthocentre of ABCABC, then PP is either the incentre or an excentre of A1B1C1A_1B_1C_1.
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