Pythagoras theorem, that the square of the longest side of a right angled triangle is equal to the sum of the squares of the other 2 sides.

Square used to prove Pythagoras theorem

This can be proved quite easily by drawing a square into which fit 4 of the same right angled triangle as shown below

As you can see the area of the whole square is equal to the the sum of the 2 shorter sides squared or (a+b)². The area of the green square left is the square of the longest side c². We also know that the area of each of the triangles is 1/2 x base x height = ab/2

From these 3 areas we can prove the theorem. The know that the total area of the square is equal to the area of the green square plus 4 of the triangles ie)

(a+b)²=c²+ 4ab/2
a² + b² + 2ab = c²+ 2ab

The 2ab ‘s cancel and we are left with Pythagoras theorem
a² + b² = c²