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.
Extend the segment $\overline {p_1p_2}$ to the chord $\overline {AB}$ ordered as $\{A,p_1,p_2,B\}$ (though we might have $p_1=A$ or $p_2=B$ or both). Suppose $d(p_1,A)≤d(p_2,B)$. Then the path $p_1\to A\to p_2$ has length no greater than the length of $\overline {AB}$ which, in turn, has length no greater than a diameter.
Note: In general, this need not be the minimal satisfactory path.