Four Golden Circles
Love this construction from Bùi Quang Tuån via Alexander Bogomolny at Cut-the-Knot.
http://www.cut-the-knot.org/do_you_know/FourGoldenCircles.shtml
Two tangent pairs of overlapping congruent circles. If the overlapping circles pass through each other's center, bam! Golden ratio all over the place.