Prisoner and tiger

An ancient Roman theater was packed with spectators. A prisoner was led to the middle of the court. Through some door at the perimeter, a hungry tiger was released. The prisoner had one chance to survice. If he succeeded to escape through the door the tiger came in, he would regain his freedom. It happens that both the tiger and the prisoner had the same speed and endurance. Question 1: Is there for the prisoner a way to escape? Can you figure out the best strategy for this prisoner? You could start with the assumption that the theater is circular and that the prisoner can be represented with points on a plane. Question 2: What happens if the theater is not circular, and what if the particpants are not points in a plane?
In this animation, you can investigate what happens if the prisoner runs along a cricular perimeter. You can place the tiger anywhere you want.