Pythagorean theorem four elementary lemmata: Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square. The proof is as follows: This proof, which appears in Euclid's "Elements" as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares. This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used. We have already discussed the Pythagorean proof, which was
Moiré pattern four sides equal to ; (we have a right triangle whose hypotenuse is and the side opposite to the angle is ). The pale lines correspond to the small diagonal of the rhombus. As the diagonals are the bisectors of the neighbouring sides, we can see that the pale line makes an angle equal to with the perpendicular of each pattern's line. Additionally, the spacing between two pale lines is , half of the long diagonal. The long diagonal is the hypotenuse of a right triangle and the sides of the right angle are and . The Pythagorean theorem gives: