Each of the images below is a link to a step-by-step GeoGebra applet through which it is possible to move forwards and backwards in the construction, or watch how the construction is solved in a short animation. The blue “circular” points in some applets are movable.
These applets are gathered in this this GeoGebra book.
regular pentagon inscribed in a circle #01
regular pentagon inscribed in a circle #02
regular pentagon inscribed in a circle #03
(adapted from the marvelous book A Geometria ao Alcance de toda a Gente (1947), authored by A. A. Ferreira de Macedo.