When you learn trigonometry, you learn a lot about the properties of triangles. But such understanding can lead to even more questions. This article will explore some interesting properties of triangles whose sides are integers.
Is there anything special about triangles with three integer sides, other than the fact that their sides are integers? Did you know that the common right triangles that have 45-degree angles or 30- and 60-degree angles cannot have three integer sides? Are only certain angles possible in triangles that have three integer sides? If so, can we predict whether or not an angle can be part of an integer triangle? How many different (non-similar) triangles can be formed which have integer sides and which all contain the same angle?
I will use an informal theorem/proof/discussion style to present some surprising insights into the nature of integer triangles. This will lead to Theorem 10, a clever formula for computing the lengths of sides of integer triangles having certain angles. We'll work on the most general kind of triangle, which has three unequal sides and three unequal angles. Such a triangle is called scalene.