A right triangle is given, from which a periodic tiling is created, from which puzzle pieces are constructed.

On three previous images, the hypotenuses of copies of the given triangle are in dashed red.  On left, a periodic square in dashed red takes another position relative to the tiling:  its center is the one of a small tile.  And one of the puzzle pieces is square, its size is the one of a small tile.  The four other puzzle pieces can form together another tile, and they are congruent, because of a rotation of a quarter turn around the center of a large tile that transforms at the same time the tiling and the grid in dashed red into themselves.  Therefore the area of a large tile equals four times the area of one of these four puzzle pieces.  In case where the initial triangle is isosceles, the midpoint of any segment in dashed red is a common vertex of four tiles with equal sizes:  a =  b,  and each puzzle piece which is a quarter of a tile is an isosceles triangle.  Whatever the shape of the initial triangle, the two assemblages of the five puzzle pieces have equal areas:
 a ² + b ²  =  c ².    Hence  the  Pythagorean  theorem.