geometry - Can squares be replaced by cycloids in the Pythagorean Theorem?

As everyone knows, for a right triangle, the square on the hypotenuse is equal to the sum of the squares on the legs.

What happens if we replace the squares with some other geometrical figure, such as one arch of a cycloid? Does the Pythagorean relation still hold? That is, is the area under one arch of a cycloid that exactly fits the hypotenuse equal to the sum of the areas under the arches of the cycloids that exactly fit the legs?

It relies on the fact that area varies as square of scaling ratio.