Subdividing triangles with $\pi$-commensurable angles

TL;DR

This paper investigates the properties of subdividing triangles with angles commensurable with pi, proving the existence of infinitely many such triangles without non-trivial subdivisions, and analyzing the role of angle bisectors in recursive subdivisions.

Contribution

It introduces the concept of pi-commensurable triangles and characterizes their subdivision properties, including the uniqueness of angle bisector subdivisions and counts of possible subdivisions.

Findings

Infinitely many pi-commensurable triangles lack non-trivial subdivisions.

Subdivision by angle bisectors is fundamental in recursive subdivisions.

Counts of pi-commensurable subdivisions and Z-degree subdivisions are provided.

Abstract

A point in the interior of a planar triangle determines a subdivision into six subtriangles. A triangle with angles commensurable with $\pi$ is called $\pi$-commensurable. For such a triangle a subdivision where each of the subtriangles are $\pi$-commensurable too is called $\pi$-commensurable. We prove that there are infinitely many $\pi$-commensurable triangles that do not admit any $\pi$-commensurable subdivision except the one given by angle bisectors. We count the number of $\pi$-commensurable subdivisions of triangles. We perform a similar count for Z-degree sub-divisions of Z-degree triangles too. Finally we show that subdivision by angle bisectors is essential in recursive subdivisions in the sense that recursive $\pi$-commensurable subdivisions of any $\pi$-commensurable triangle ultimately involve a subdivision by angle bisectors.