Area of big square = (b + a)^2
       =    (area of square 5) + (area of triangle 1)
         + (area of triangle 2) + (area of triangle 3)
          + (area of triangle 4)                                       1.a
 
area of square 5 = c^2                                                                          1.b
area of triangle = (1/2)b*a                                                                     1.c

Therefore, substituting 1.b and 1.c  into 1.a and using the fact 
that all four triangles are equal gives;

b^2 + 2a*b +a^ = c^2 + 4(b*a/2)
                            = c^2 + 2a*b.

Cancelling the 2a*b gives the desired result,

 a^ + b^2 = c^2

side a

side b

side c

1

2

3

4

5

In this proof of the Pythagorean thereom, we have square 5 
which is circumscribe in a bigger square.  The four triangles
are all equal in size.

I learn about this proof when I was in grade school.  This is 
how the Chinese proved it more than 2000 years before
Pythagoras.

For more infornmation go to the link provided below:
http://www.hitxp.com/math/geo/euclid/051202.htm

Proof of Pythagorean thereom

A = (-1.1882802547770699, 3.6876470588235293)
B = (-1.1682802547770699, 2.2876470588235294)
poly1, a, b, c, d, C, D = Polygon(A, B, 4.0)
E = Point(d)
F = Point(a)
poly2, e, f, g, h, G, H = Polygon(E, F, 4.0)