Reflected circles, and congruent perspective triangles

Abstract

For any three points X, Y, Z, let ⊙XYZ denote the circle through X, Y, Z (the circumcircle of ∆XYZ) or, if X, Y, Z happen to be collinear, the line XYZ. (We shall often regard lines as special circles, circles of infinite radius.) This paper is about the following theorem, and extensions of it:Theorem 1: Given ∆ABC and a point P, reflect ⊙PBC, ⊙APC, ⊙ABP in the lines BC, AC, AB respectively. Then the three reflected circles have a common point, Q (see Figure 1).I do not know if this theorem is new, but I have not come across it in the literature. The reader is invited to prove it by angle-chasing, using circle theorems: let two of the reflected circles meet at Q and then prove that this point lies on the third reflected circle. This method is rather diagram-dependent, and does not seem to lead to the extensions of Theorem 1 referred to above in any very obvious manner, so we shall adopt a different approach.