Given a circle, inscribe a regular pentagon.
Although we generally consider the circle as given, in theory we should always start with two given (arbitrarily chosen) points.
Given points A and O, draw circle O with radius AO and diameter AA'. Let AO be 1 unit.
Construct the perpendicular bisector of AA' and find one intersection with circle O; call this S.
Construct the perpendicular bisector of SO and find its intersection with line SO; call this M.
Draw a circle M with radius OM.
Draw a line through A' and M. Find its intersections with circle M; call these P and Q.
Draw (two) circles with center A' and radii QA' and PA'. Find their intersections with circle O; call these B, C, D, and E.
Draw lines AB, BC, CD, DE, and EA.
I say the figure ABCDE is a regular pentagon. QEF.