First of all, I'm sorry for just describing without a drawing - I'm having problems with that.
be a line segment connecting a point on the regular
-gon to another, which is k vertices far (by symmetry all of those are congruent). That is, there are k vertices between the two connected vertices. By drawing the circumscribed circle of the regular polygon, it can be seen that the angle between two line segments ending in two adjacent vertices and starting at another vertex is
. Draw two consecutive
for some k. The angle between them is, as said,
. Also, the angle between
and the side of the polygon inside the triangle formed is
(this too can be seen by drawing the circumscribed circle and counting equal arcs). In either case, from the sinus law in the triangle formed (by
and a side of the polygon) we get
s are equal in some order, the answers are
Is there a way to proceed with this expression?
Explanation to the last step:
First we derive a formula for
yields the last expression for the sum of distances above.