Explanation

Assume b > a, so that the lines parallel and perpendicular to the hypotenuse of ABC are drawn in the square ACB_cB_a through its midpoint M. The length of both segments of these lines inside the square equal the length of the hypotenuse c. Each segment is divided by M into equal parts of length c/2.

Drop a perpendicular from M onto CB_c. Its foot will land in the middle of CB_c. Thus a triangle will be formed with the sides parallel to those of ABC, but just half as big.

From here we conclude that the endpoints on the add-on lines in square ACB_cB_a divide its sides (each of length b) into two parts (a + b)/2 and (b - a)/2. The difference of the two

which explains the emergence of a square of side a after rearrangement of the pieces on the hypotenuse.

Equidecomposition by Dissection