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arXiv · 2206.04348

Subdividing triangles with $\pi$-commensurable angles

Hasan Korkmaz·Ferit Öztürk
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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.

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Hasan Korkmaz, Ferit Öztürk. 2022-06-09. Subdividing triangles with $\pi$-commensurable angles. https://arxiv.org/abs/2206.04348

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