The altitude to the hypotenuse forms three similar triangles:

Notice that the similarities must be named consistently.  In the above, the letters are in the order of double-marked angle, right angle, single-marked angle.

These similarities lead to important proportions, which can be more easily represented if we label the sides of the triangles:  In , let a be the length of the side opposite angle A, b be the length of the side opposite angle B, and c be the length of the hypotenuse.  Then in , b is the length of the hypotenuse; let d be the length of  and h the length of , which is the altitude of .  In , the length of the hypotenuse is a and the length of  is h; let e be the length of .  Notice that d + e = c. 

As an example, consider .  If we consider the hypotenuse and short leg of each, the following ratios of corresponding parts are equal:  .  If we cross-multiply, we get .  This says the product of the hypotenuse and the altitude of a right triangle (in this case, ) is equal to the product of its legs.