Prove that a string cant be outside a circle How can I prove that a chord can't be outside the circle itself.
Is there a way to prove that you can't draw a chord outside the circle.

We'll give a solution that works as well in hyperbolic geometry and perhaps in some other geometries.
Let $O$ the center of the circle, $A$, $B$ distinct points on the circle and $M$ a point on the segment $AB$. Consider the angles $\angle OMA  $ and $\angle OMB  $ summing up to $\pi$. Therefore, one of them has size $\ge \pi/2$. Assume it's $\angle OMA \ge \pi/2  $.  In the triangle $\Delta OMA$ the sum of the angles is $\ge \pi$. Therefore $\angle OAM < \pi/2 \le \angle OMA$ and therefore for the opposite sides we have $OM < OA$. 
