Nick Lord says that this problem is
immediately enticing for students (and for him) and there are
many possible approaches to explore and compare.
In the equilateral triangle inscribed in the circle, X and Y are
the midpoints of the edges. Find the ratio
.
The square sits on a diameter of the circle. Find the ratio
. The regular pentagon is inscribed in the circle. Find the ratio
.
See also the problem
Gold Again.