If a straight line be cut in extreme and mean ratio, and there be added to it a straight line equal to the greater segment, the whole straight line is cut in extreme and mean ratio, and the original straight line is the greater segment.
Proof
Extending by the greater segment AC to A′ (so A′A=AC, AB
the original), check that A′A:AB=AB:(A′A+AB−AC), which
reduces via the original extreme-and-mean relation to the same form.