[escherplex]

The intuitive geometric description I sometimes use with people is to have them imagine two duplicate rectangles and cut a diagonal in each to form four equivalent triangles. Mentally label the hypotenuse c, the long side b, and the short side a. Then arrange the four triangles into a square with the perimeter consisting of sides of length a+b. Now from the outside area is simply (a+b)^2. But computed from the inside area will consist of 4 * (1/2 a*b) + the area of the central tilted square with sides c or c^2. Simple algebra yields a^2 + 2ab + b^2 = 2ab + c^2 or a^2 + b^2 = c^2. A more fun problem is imagining why there are 180 degrees in any triangle. And for that matter where did that 360 degrees in a circle come from (last one hint: imagine 6 duplicate equilateral triangles and arrange them into a hexagon, assign the Babylonian magic number 60 to each vertex and sum the interior central angles, after using one side to draw a circle around the hexagon).