Hirano's Construction of a Regular Pentagon
This Demonstration shows Yosifusa Hirano's construction of a regular pentagon with a ruler and compass.
Proof that the constructed figure is a pentagon:
Let circle be a unit circle with center with and , since is the midpoint of . Then by the Pythagorean theorem is . Since is , we have . and angle is an inscribed angle that subtends a semicircle, thus angle is a right angle. Letting be angle , , and thus . Then by the central angles theorem, angle , making angle , which proves that is a pentagon.
 A. Bogomolny. "Regular Pentagon Construction by Y. Hirano." Interactive Mathematics Miscellany and Puzzles. (Sept 8, 2014) www.cut-the-knot.org/pythagoras/PentagonConstruction.shtml.