A Special Conic Associated with the Reuleaux Negative Pedal Curve

TL;DR

This paper investigates a special conic related to the Reuleaux negative pedal curve, revealing unique geometric properties and providing a synthetic proof using classical geometric techniques.

Contribution

It introduces a novel conic associated with the Reuleaux negative pedal curve and proves its properties using synthetic geometric methods.

Findings

The conic passes through four arc endpoints and a specific point P0.

One focus of the conic is the boundary point M.

The paper provides a synthetic proof based on Poncelet's polar duality and inversive techniques.

Abstract

The Negative Pedal Curve of the Reuleaux Triangle w.r. to a point $M$ on its boundary consists of two elliptic arcs and a point $P_0$. Interestingly, the conic passing through the four arc endpoints and by $P_0$ has a remarkable property: one of its foci is $M$. We provide a synthetic proof based on Poncelet's polar duality and inversive techniques. Additional intriguing properties of Reuleaux negative pedal are proved using straightforward techniques.