This graphical 'proof' of the Pythagorean Theorem starts with the right triangle below, which has sides of length a, b and c. It demonstrates that a2 + b2 = c2, which is the Pythagorean Theorem.
It is not strictly a proof, since it does not prove every step (for example it does not prove that the empty squares really are squares). But it does demonstrate the theorem in an interesting way.
|Make 3 copies of the original triangle and arrange the four triangles in a square as shown. The outer square JKLM will remain fixed throughout the rest of the proof.
|Each side of the empty square in the middle has a length of c, and so has an area of c2.
|Re-arrange the triangles as shown so that the empty space is now divided into two smaller squares.
|Notice that the top left empty square has each side equal to a, so its area is a2.
|Notice also that the bottom right empty square has each side equal to b,
so its area is b2.
|Done. We have rearranged the triangles inside a constant-size square.
The empty space we started with ( c2 ) must be equal to the sum of the two empty spaces at the end.
Therefore a2+b2 = c2 QED.