Construct a pentagon/pentagram
Problem: Inscribed Pentagram
Given a circle, inscribe a regular pentagram.
A pentagram is also known as a stellated pentagon. This is only a nice way of saying star-shaped. It has the same construction as the pentagon up to the final step; but instead of connecting adjacent vertices, connect every other. Given the pentagon, the pentagram is trivial. Together with its circumscribing circle, this is known as a pentacle.
This symbol has a long and diverse heritage; the ancient Greeks and Babylonians, pagans in all ages, and early Christians all found use for it. Japanese anime commonly uses it to indicate magic; and it's used in the flags of Morocco and Ethiopia. Sir Gawain, a Knight of King Arthur's Round Table, bore a pentagram on his shield symbolizing the five senses, five fingers, five joys of Mary, five wounds of Christ, and the five virtues of knighthood. And Goethe's Faust imprisons the demon Mephistopheles in a pentagram.