Suppose there are two points, $p_1$ and $p_2$, inside a circle of radius $R$. You must travel from $p_1$ to $p_2$ but you must first "touch" a point on the circle before arriving at $p_2$. Assuming you always use the shortest path possible, can your path ever be longer than $2R$?
After trying a few examples it appears the answer is NO. But I'm finding it tough to prove this. Any help appreciated.