Garfield’s Trapezium

James Abram Garfield was a member of the United States House of Representatives when he submitted a proof of the Pythagorean Theorem to the New England Journal of Education in 1876. He of course went on to become the President of the United States. His proof (shown below) was unusual in that it used a trapezium (trapezoid for US readers).

garfield trap1

It’s a straightforward proof to follow, and can be used quite easily in the classroom with a little guidance. But a trapezium structured like this offers so much more than just (just!) a proof of the Pythagorean Theorem.

Take the following configuration, which also shows the enclosing rectangle:


This allows us to investigate angles and determine some properties of the inverse tangent function, arcTan:


In fact you can use Garfield’s Trapezium to derive a whole host of trigonometric functions. Take the example below, which enables you to fairly easily find the tricky trig functions of 15 and 75 degrees:


Fun! As a challenge to you, can you use Garfield’s Trapezium to derive the addition and subtraction formulas for Tan?