1. By: Mark Nakanishi Source: Cut-the-knot.com
Pythagorean Theorem
By: Mark Nakanishi
Source: Cut-the-knot.com

2. Step One We start with four copies of the same right triangle.
Each has area ab/2, since the area of a triangle is width times height.

3. Step Two
                        
Three of the triangles have been rotated 90o, 180o, and 270o, respectively.
Then we put them together without additional rotations so that they form a square with side c.

4. Step Three The square has a square hole with the side (a-b).
                        
The square has a square hole with the side (a-b).
So the square hole =(a-b)2
The area of the four triangles = (4·ab/2), which can be simplified into 2ab.
So the area of the whole square =(a-b)2+2ab

5. Step Four Since the big square equals c2: c2 = (a-b)2+2ab
                        
Since the big square equals c2: c2 = (a-b)2+2ab
(a-b)2+2ab can be simplified to a2-2ab+b2+2ab
With cancellation a2-2ab+b2+2ab = c2 can be simplified to…………

6. Which proves our Pythagorean theorem.
a2+b2 = c2 !!!!!
Which proves our Pythagorean theorem.