Please list two proofs of Pythagorean theorem.

Let the sides of a triangle facing the three vertices be A, B and C, where C is the hypotenuse. Draw a high line on C, and divide C into two parts, X and Y, where X is near vertex A and Y is near B. By using the projective theorem (the projective theorem comes from triangle similarity, and the proof of triangle similarity does not need Pythagorean theorem, so it can be used as proof of Pythagorean theorem), we can get the square of A =cy, and B = CX+cy = C(x+y)= C, and draw the conclusion.

Area method:

As shown in the figure, the area of ABDE is equal to the square of C and the sum of the areas of four right triangles and small squares. The area of four right-angled triangles is half of ab times 4=2ab, and the side length of the small square is (b-a).

So the sum of the areas is 2ab+(b-a) 2 = A2+B2-2ab+2ab = the square of a+the square of b,

That is, the square of a+the square of b = the square of c.