Albion College

Mathematics and Computer Science

Mathematics and Computer Science

COLLOQUIUM

Grade School Triangles and Ailles' Rectangle

Jack Calcut

In grade school, students learn a standard set of Euclidean
triangles. Among this set, the usual 45-45-90 and 30-60-90
triangles are the only right triangles with rational angles and side
lengths each containing at most one square root. Are there any other
such right triangles? We answer this question and present an elegant
complement, called Ailles' rectangle, that deserves to be in every
geometry teacher's toolkit.