Pythagorean Theorem

 
Start with a right triangle.
C=90.
Since A+B+C =180,
A+B = 90.
This fact lets us make squares and rectangles out of this triangle.
Two of these triangles make an a x b rectangle.

 
Triangles arranged to make
a square of sides a+b.
This square of sides (a+b) is made from a
  c x c square
and 4 triangles which can be rearranged to 
form 2 a x b rectangles.
So (a+b)^2 = c^2 + 2ab.
Recall (a+b)^2 = a^2 + 2ab + b^2
So
 a^2 + 2ab + b^2 = c^2 + 2ab 
or
a^2 + b^2 = c^2

And another way to do it . . .

Arranged this way, the triangles form a right angle. One sticks out a length of b-a
A square of sides c.
c^2 = (b-a)^2 +2ab
Recall (b-a)^2 = a^2 - 2ab + b^2
So
c^2 = a^2 - 2 ab + b^2  + 2ab
c^2 = a^2 + b^2

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