In a given circle to inscribe a fifteen-angled figure which shall be both equilateral and equiangular.
Proof
Inscribe a regular pentagon (IV.11) and a regular equilateral
triangle (IV.2) in the circle, sharing a common vertex A. The arc
from A to the next pentagon-vertex is 51 of the circle;
the arc from A to the next triangle-vertex is 31. The
difference is 31−51=152 of the
circle. Bisect that arc (III.30); each half is 151 of
the circle, and stepping that chord fifteen times around gives the
regular pentadecagon.