Given a circle, inscribe a regular pentagon.
Beginning geometers quickly learn how to inscribe a number of polygons within a given circle. Construction of a hexagon (6 sides) is almost automatic; this leads directly to the (inscribed) equilateral triangle. With the perpendicular bisector comes the square, and so to the octagon (8 sides). A little concentration on these techniques allows one to construct as well the dodecagon (12 sides). Thus constructions are easily found for regular polygons with sides 3,4,6,8, and 12. Conspicuously missing are 5,7,9, 10, and 11.
However, it is impossible to construct a regular heptagon (7 sides), nonagon (9 sides), or hendecagon (11 sides). Draftsmen may use a variety of techniques to approximate these figures but we are interested here only in exact constructions using the ideal compass and straightedge. Thus the outstanding problems are the pentagon (5 sides) and the decagon (10 sides). A little thought shows that if we are able to construct either one, the other will come easily.
Relative to the other polygons, the construction of the pentagon is quite complex and non-intuitive. This must have appeared quite mysterious to the ancients and it is of no surprise that those able to perform the feat were considered magicians and sorcerers. Here we will dispense with superstition and introduce an elegant construction with a proof suitable for beginners.