How do you prove the law of cosines using the geometry of a triangle?

To prove the law of cosines, consider a triangle ABC with sides a, b, and c opposite angles A, B, and C, respectively. Drop a perpendicular from C to side AB, dividing it into segments of lengths x and (b-x). Using the Pythagorean theorem in the resulting right triangles, express c² in terms of a, b, and cos(C).