Seeing Pythagoras

By 
John Diamantopoulos and the Plus Team

The great thing about geometry is that you can sometimes prove things using pictures. A great example is Pythagoras’ famous theorem about right-angled triangles. Writing $a$ and $b$ for the two shorter sides and $c$ for the side opposite the right angle, we always have

  \[ a^2+b^2=c^2. \]    

Below are three visual proofs of Pythagoras' theorem, which were sent to Plus by John Diamantopoulos, Professor of Mathematics at Northeastern State University.

The first visual proof is probably similar to the one Pythagoras himself used. To replay the animated gif simply hover your mouse over the image.

Visual proof 1

The second visual proof is probably similar to one used by the ancient Indian mathematician Bhaskara. To replay the animated gif simply hover your mouse over the image.

Visual proof 2

The third visual proof was originally constructed by the 20th President of the United States, James Garfield. We do wonder whether the current US president also dabbles in geometry in his spare time. To replay the animated gif simply hover your mouse over the image.

Visual proof 3

As you’ll have noticed, the last visual proof uses the fact that

  \[ (a+b)^2=a^2+2ab+b^2. \]    

This too can be demonstrated using pictures. To replay the animated gif simply hover your mouse over the image.

Visual proof 1

About this article

John C. D. Diamantopoulos

John C. D. Diamantopoulos is a Professor in the Department of Mathematics and Computer Science at Northeastern State University in Tahlequah, Oklahoma. His mathematical interests include ordinary differential equations, mathematics education and the history of mathematics. Diamantopoulos is also very active in his church, volunteering on computer productions/presentations and any area that needs attention.