How did Euclid prove the Pythagorean theorem?

In his seminal work, the Elements, Euclid provided a famous demonstration of the Pythagorean theorem known as the "Windmill proof" due to the shape of the geometric figure used.1

The proof relies on constructing squares on each side of a right triangle. By drawing a line from the right angle to the opposite side of the square on the hypotenuse, Euclid divided that square into two rectangles. He then demonstrated that each rectangle is equal in area to one of the squares built on the triangle's legs. This was achieved by showing that the triangles formed within the construction are congruent and by applying theorems regarding the areas of triangles and parallelograms.1

Specifically, Euclid proved that the area of the square on each leg is equal to the area of the corresponding rectangle within the square on the hypotenuse. Consequently, the sum of the areas of the two smaller squares equals the area of the square on the hypotenuse, confirming the relationship $a^2+b^2=c^2$.1