Inscribing Regular Pentagon in Circle
In the words of Euclid:
Let $ABCDE$ be the given circle (although note that at this stage the positions of the points $A, B, C, D, E$ have not been established).
Let $\triangle FGH$ be constructed such that $\angle FGH = \angle FHG = 2 \angle GFH$.
Let $ACD$ be inscribed in $ABCDE$ such that $\angle ACD = \angle FGH, \angle ADC = \angle FHG, \angle CAD = \angle GFH$.
We have that $\angle CDA = \angle DCA = 2 \angle CAD$.
As $\angle CDA$ and $ \angle DCA$ have been bisected, $\angle DAC = \angle ACE = \angle ECD = \angle CDB = \angle BDA$.
Hence from Equal Arcs of Circles Subtended by Equal Straight Lines, the straight lines $AB, BC, CD, DE, EA$ are all equal.
So from Angles on Equal Arcs are Equal $\angle BAE = \angle AED$.
For the same reason, $\angle BAE = \angle AED = \angle ABC = \angle BCD = \angle CDE$.