**Problem IX:** Given a straight line AB, to extend it to G so that the total AG to the extreme GB has a given ratio C to D.

Construction: Draw from A a line AE making any angle with AB, and put on it AF equal to C and EF equal to D. Join FB. I say that if a parallel EG is drawn to that, intersecting AB extended at G, then AG is to GB as AE is to EF, that is, C to D.