A Singular Miquel Configuration and the Miquel Direct Similarity

TL;DR

This paper explores a special Miquel configuration involving a triangle and its associated circles, revealing collinearity properties and analyzing the direct similarity between the original triangle and a triangle formed by Miquel circle centers.

Contribution

It introduces a specific Miquel configuration with new collinearity results and investigates the direct similarity between the original and Miquel circle center triangles.

Findings

ASQ and BSR are straight lines in the configuration

Identifies a direct similarity between the original triangle and the Miquel circle centers triangle

Provides geometric proofs for collinearity and similarity properties

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.