First of all, I'm sorry for just describing without a drawing - I'm having problems with that.
Let the
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
s:
and
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
or
(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
.
Because the
s and
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
:
Therefore:
And thus
Subtituting in
and using
yields the last expression for the sum of distances above.