The Six-Point Circle Theorem

TL;DR

This paper investigates six special pedal triangles within a given triangle, showing that their six pedal points all lie on a common circle, revealing a new geometric circle theorem.

Contribution

It establishes the existence and uniqueness of six minimal-area pedal triangles with prescribed angles and proves that their pedal points are concyclic.

Findings

Six minimal-area pedal triangles exist for given angles.

Each pedal triangle corresponds to a unique pedal point.

All six pedal points lie on a single circle.

Abstract

Given $\Delta ABC$ and angles $\alpha,\beta,\gamma\in(0,\pi)$ with $\alpha+\beta+\gamma=\pi$, we study the properties of the triangle $DEF$ which satisfies: (i) $D\in BC$, $E\in AC$, $F\in AB$, (ii) $\aangle D=\alpha$, $\aangle E=\beta$, $\aangle F=\gamma$, (iii) $\Delta DEF$ has the minimal area in the class of triangles satisfying (i) and (ii). In particular, we show that minimizer $\Delta DEF$, exists, is unique and is a pedal triangle, corresponding to a certain pedal point $P$. Permuting the roles played by the angles $\alpha,\beta,\gamma$ in (ii), yields a total of six such area-minimizing triangles, which are pedal relative to six pedal points, say, $P_1,....,P_6$. The main result of the paper is the fact that there exists a circle which contains all six points.