On a Conjecture on Sharygin Triangles

TL;DR

This paper proves a conjecture stating that, among regular polygons inscribed in a unit circle, there is only one Sharygin triangle (up to isometry) formed by vertices of the polygon.

Contribution

The paper introduces a simple method to prove the uniqueness of the Sharygin triangle among vertices of regular polygons inscribed in a circle.

Findings

Only one Sharygin triangle exists among vertices of a regular polygon on a unit circle.

The proof confirms the conjecture for all regular polygons inscribed in the circle.

The method simplifies previous approaches to this geometric problem.

Abstract

By a simple method we prove the following conjecture on Sharygin triangles: there is only one Sharygin triangle (up to an isometry) whose vertices are chosen from the set of vertices of a regular polygon inscribed in a circle of radius 1.