Details

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.

Reference

[1] A. Bogomolny. "Regular Pentagon Construction by Y. Hirano." Interactive Mathematics Miscellany and Puzzles. (Sept 8, 2014) www.cut-the-knot.org/pythagoras/PentagonConstruction.shtml.