These two propositions are usually proved by the help of the Pythagorean Theorem. Some writers, however, actually construct the squares and give a proof similar to the one in that proposition. This plan goes back at least to Gregoire de St. Vincent (1647).
It should be observed that
a2 = b2 + c2 - 2b'c.
If ∠A = 90°, then b' = 0, and this becomes
a2 = b2 + c2.
If ∠A is obtuse, then b' passes through 0 and becomes negative, and a2 = b2 + c2 + 2b'c.
Thus we have three propositions in one.
