A New Equilateral Triangle?

TL;DR

This paper explores a geometric iterative process involving triangle bisectors and circumcircles, revealing that repeated application converges to a pair of equilateral triangles aligned with Morley's triangle.

Contribution

It introduces a novel iterative construction based on bisectors and circumcircles, demonstrating convergence to equilateral triangles related to Morley's theorem.

Findings

Iterative process converges to equilateral triangles.

The resulting triangles are parallel to Morley's triangle.

The process reveals new geometric relationships.

Abstract

In any triangle, the perpendicular side bisectors meet the corresponding internal angle bisectors on the circumcircle. If we take those three points as the vertices of a new triangle and repeat the operation indefinitly, we end up in the limit with a par of equilateral triangles whose sides are parallel to the sides of the Morley triangle of the initial triangle.