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.