Squares in triangles

Abstract

When can a triangle cover a square? More precisely, when can a square of side s fit into a triangle with sides a, b, c?Questions about when, and whether, one shape can fit into another are fundamental to the elementary study of shapes in geometry, and yet such questions have only rarely been considered in the literature. In 1872 Cayley thought the question of determining precisely when a given triangle fits into a given circle sufficiently interesting that he posed it as a problem in the Educational Times. In 1956 Ford asked when one given rectangle fits in another given rectangle. A necessary and sufficient condition on the sides was soon supplied by Carver (see also Wetzel). The corresponding question for two triangles, asked in 1964 by Steinhaus, went unanswered for thirty years until Post supplied a set of 18 inequalities on the six sides whose disjunction is both necessary and sufficient.