Incommensurability of the Base and Leg in an Isosceles Triangle

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.)

[more]

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.

[less]

Contributed by: Izidor Hafner (September 2012)
Open content licensed under CC BY-NC-SA


Snapshots


Details



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send