In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle.
Proof
By VI.8 the altitude from the right angle cuts the hypotenuse into
segments such that each leg is the mean proportional between the
hypotenuse and the adjacent segment. Applying VI.20 (areas of
similar figures are in the duplicate ratio of corresponding sides)
gives each leg-figure equal to its adjacent piece of the
hypotenuse-figure. Summing the two pieces by V.24 yields the
hypotenuse-figure.