Problem: Build a regular pentagon knowing one segment *PQ* whose endpoints are one of the vertices of the pentagon and the midpoint of the opposite side.

Construction: Let *BCDEF* be any regular pentagon. Draw the line segment *BG* that joins the vertex *B* with the midpoint of the side *ED*.

*If * If *BG = PQ*, then the regular pentagon *BCDEF* is the solution to the problem. If not, the side of the pentagon solution (say *z*) will be the fourth proportional of the segments *BG*, *PQ* and *ED*. That is,

*BG*:*PQ = ED*:*z*.