In an isosceles triangle, if the measure of the leg angle is 72°, then the base and a leg are incommensurable. (Two lengths
are called incommensurable if their ratio is irrational.)
Let the vertices of the triangle be
is the base. The angles are 72°, 72°, and 36°.
Assume that the ratio of
is rational, say
for positive integers
. For some appropriate unit of measurement, we can take the lengths of
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
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.