Problem: Let RSTUV be a given polygon and PQ a given line segment. Construct a polygon MNKIL similar to the previous and equally arranged so that if segment MN, homologous to RS, is taken from LN, homologous to VS, and to the rest you add segment LI, homologous to VU, you obtain a segment equal to PQ.
Construction: Let BCDEF be any polygon similar to RSTUV and equally arranged. On segment CF take point G so that CG = CB. Extend segment CF to point H so that FH = FE.
If HG = PQ, then the polygon BCDEF is the solution to the problem. If not, we will determine the fourth proportional (say w) with respect to the segments HG, PQ and ED. Then, w = IK will be the segment homologous to ED in the solution polygon. From it the required polygon will be constructed.