Proportional segments in a right triangle

In a right triangle ABC, if we drop the altitude CD from the right-angle vertex to the hypotenuse, the triangles ABC, ACD and CBD are similar to each other because each pair has two equal angles. For segments a, b, c, if a : b = b : c then b is the mean proportional between a and c, that is b² = ac. From the similarity: the altitude h is the mean proportional between the projections a_c and b_c of the legs on the hypotenuse, so h² = a_c·b_c. Each leg is the mean proportional between the hypotenuse and its own projection: a² = c·a_c and b² = c·b_c. Adding these gives a² + b² = c·(a_c + b_c) = c², which is the Pythagorean theorem.