Problem 429: Circumscribed and Inscribed Regular Pentagon, Perpendicular, Measurement. Level: High School, College, SAT Prep.

In the figure, a is the side of a regular pentagon ABCDE circumscribed to circle O and b is the side of a regular pentagon A'B'C'D'E' inscribed in circle O. Lines A'A''B', B'B''C'', C'C''D'', D'D''E'', and E'E''A'' are perpendicular to A'E', A'B', B'C', C'D', and D'E', respectively. If c is the side of the pentagon A''B''C''D''E'', prove that .