Poristic virtues of a negative pedal curve

TL;DR

This paper explores the geometric properties of triangles sharing certain circles and introduces a special negative-pedal curve that reveals poristic relationships, leading to new geometric configurations.

Contribution

It identifies a unique negative-pedal curve related to pedal circles and demonstrates its role in establishing poristic triangle pairs.

Findings

Negative-pedal curve is a special i-conic related to pedal circles.

Poristic pairs emerge from the geometric configurations.

The study reveals new poristic relationships in triangle geometry.

Abstract

We describe all triangles that shares either circumcircle and pedal circle or circumcircle and negative-pedal circle. Neither of these pairs is poristic; nevertheless, the negative-pedal curve of the pedal-circle is a (very) special i-conic that points toward a poristic solution. Subsequently, other poristic pairs show up and the choreography swiftly begins.