Mathematicians prove theorems. Once a theorem is proved, it is true for all time. The theorems proved by the ancient Greeks are as true today as they were over two thousand years ago, and the theorems proved today will be true even if, after millions of years, humans evolve into another species. In this chapter we present some delightful and sometimes surprising theorems.

A Square inside Every Triangle

Given any triangle, is it always possible to inscribe a square in it? We require that the square has a side on one of the sides of the triangle, with the other two corners touching the other sides of the triangle.

The answer is yes, by similarity. Put the triangle on top of a square, as ΔABC is placed in Figure 4.1. Now extend the other two sides of ΔABC so that they meet the line that the square sits on. This results in a triangle similar to the given triangle and circumscribing the square. Finally, change the scale of the whole diagram so that the circumscribing triangle is the same size as our given triangle—and we are done. Note that the side of the triangle we place on the square must be chosen so that the altitude to that side lies inside the triangle. We have proved the theorem.