Halving Triangular Pancakes

SummaryWe provide a method for explicitly finding the line that simultaneously halves the areas of two triangles; such a line is guaranteed to exist by the Pancake Theorem and is unique if the triangles are not overlapping. We also address the question of whether this halving line is always constructible with straight-edge and compass. This was answered negatively by Berele and Catoiu in a 2024 paper, where they provided an example of two non-congruent right-angle triangles whose halving line is non-constructible. Our approach is different from that of Berele and Catoiu, treating the problem of constructibility of the common tangent lines of two hyperbolas. In particular, we obtain an example of two congruent right-angle triangles with non-constructible halving line. We also point out some relatively unexpected cases where the halving line is constructible. Some remarks on the uniqueness and non-uniqueness of the halving line for a pair of overlapping triangles are also made.