A Singular Miquel Configuration and the Miquel Direct Similarity

Abstract

Let ABC be a triangle with P on AB, and let circle APC meet BC at Q and circle BPC meet CA at R, then the special Miquel configuration is when P, Q, R are the operative points on the sides. We show that in this case if S is the Miquel point then ASQ and BSR are straight lines. In the last part we investigate the direct similarity between ABC and DEF the triangle formed by the centres of the Miquel circles.