Points, whose pedal triangles are similar to the given triangle

TL;DR

This paper investigates eleven special points related to a given triangle, showing that six with positively oriented pedal triangles lie on a circle, while five with negatively oriented pedal triangles are collinear.

Contribution

It identifies and characterizes the geometric loci of points with pedal triangles similar to a given triangle, revealing a circle and a line for different orientation cases.

Findings

Six points with positively oriented pedal triangles lie on a circle.

Five points with negatively oriented pedal triangles are collinear.

The paper provides a geometric proof of these loci.

Abstract

We study the eleven points in the plane of a given triangle, whose pedal triangles are similar to the given one. We prove that the six points whose pedal triangles are positively oriented, lie on a single circle, while the five points, whose pedal triangles are negatively oriented, lie on a common straight line.