Remark 2: Euclid's proof amounts to solving the above equation. Construct the square ABCD, given segment AB. Let E be the midpoint of AD. Construct the point of intersection, F, of the dircde centered at E of radius EB and the extension of segments DA, as shown. Now construct square AFGX, all of whose sides have length AF, with X on segment AB.
Claim: X is the desired point.
Apply the Pythagorean Theorem to the right triangle ABE to find EB in terms of a.