Incommensurability of the Base and Leg in an Isosceles Triangle

In an isosceles triangle, if the measure of the leg angle is 72°, then the base and a leg are incommensurable. (Two lengths and are called incommensurable if their ratio is irrational.)

Let the vertices of the triangle be , where is the base. The angles are 72°, 72°, and 36°.

Assume that the ratio of to is rational, say for positive integers and . For some appropriate unit of measurement, we can take the lengths of and to be and , where .

Bisect the angle at to meet the opposite side at . The triangle is an isosceles triangle because the angles are 36°, 36°, and 108°. The triangle is an isosceles triangle because its angles are 72°, 72°, and 36° ( is similar to the original triangle ). Therefore, and .

Iterating this construction gives the points , , … and , , … with , , …, where each of the integers is positive and forms a strictly decreasing sequence, which is impossible.

Therefore, the ratio of the sides is irrational.

In fact, the ratio is , the golden ratio. Its continued fraction is with cumulants 1, 2, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, …; you can see the numerators and denominators in the table.