I was fascinated by this problem from the first moment I saw it.
Let $P$ be a convex polygon which has no two sides which are parallel. Each side $A_iA_{i+1}$ has a furthest away point $C_i$. Prove that $ \sum_{i=1}^n \angle A_iC_iA_{i+1}=\pi$
This was given in 2005 at a contest for junior level olympiad students in Romania.
I've just compiled and understood a solution, which is quite complicated (you can see it here)
Is there a simple solution for this problem using some kind of geometric transformation, or clever reduction to a particular case? (Obviously, the case of a cyclic polygon is very easy)
At first sight it didn't seem intuitive for me that for any polygon that sum of angles should be constant. Maybe some deeper insight into the problem or a heuristical argument could explain the result better...