In this construction, the points used are shown in alphabetical order. The colored golden rectangle is rectangle BKOF, and the golden triangle (shown in orange) is triangle FPB. There are two golden gnomons, shown in blue: triangle QFP, and triangle PBR. The regular pentagon is BRPQF. Every circle, line, ray, and segment used, even just to bisect segments, is shown — nothing has been hidden. This construction works because the long-edge-to-short-edge ratio of the golden rectangle is the golden ratio — and so is the diagonal-to-side ratio for the regular pentagon.

I used Geometer’s Sketchpad to make this, but everything shown can be done with the traditional Euclidean construction tools: a compass, and an unmarked straightedge.