Subdividing triangles with $π$-commensurable angles

A point in the interior of a planar triangle determines a subdivision into six subtriangles. A triangle with angles commensurable with $π$ is called $π$-commensurable. For such a triangle a subdivision where each of the subtriangles are $π$-commensurable too is called $π$-commensurable. We prove that there are infinitely many $π$-commensurable triangles that do not admit any $π$-commensurable subdivision except the one given by angle bisectors. We count the number of $π$-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 $π$-commensurable subdivisions of any $π$-commensurable triangle ultimately involve a subdivision by angle bisectors.

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