Miquel point and isogonal conjugation

TL;DR

This paper investigates the positions of the Miquel point related to a given triangle, identifying eleven such positions, and explores the conditions under which Miquel triangles are similar or isogonally conjugate to the original triangle.

Contribution

It provides a complete classification of the eleven possible positions of the Miquel point and analyzes the isogonal conjugation relations among similar Miquel triangles.

Findings

Identified eleven possible positions of the Miquel point.

Characterized when Miquel triangles are similar to the original.

Determined which similar Miquel triangles are isogonally conjugate.

Abstract

We study the possible positions of the Miquel point in the plane of a given triangle, which Miquel triangles are similar to the given one. We found out that these positions are eleven. We also study the possible positions of the Miquel point in the plane of a given triangle, where among the families of Miquel triangles there are triangles, which are similar to the given triangle. We study which of them are isogonal conjugated.