Frankly I cannot visualize the problem. If you can, please attach a diagram.
You are a chicken that cannot fly in the center of a circular chicken coop.
Farmer joe wants to eat you, but he is lazy and will only catch you if you are at some point on the circumference. Also, he moves at a rate 4 times faster than you along the circumference towards the point you are headed for.
If you get to the circumference before the farmer you can trigger an electric fence and he will die. If he gets to the circumference before you he will catch you and eat you.
Is there any point on the circumference that you can get to before farmer joe?
Hint: Angular velocity.
So we are in the center.
Whichever direction we move, the farmer moves around the circumference in that direction. Assume he takes the shortest path possible.
You are allowed to change strategies at anytime, in which case the farmer does also. He moves along the circumference towards whichever direction you are headed.
When you start from center and reach the circumference,you are traveling a distance R in time t. During that time the farmer moves through the distance 4R towards you. That means, with respect to the center , he moves through an angle 4 radians. So to escape from him you have move in a direction, which is greater than 4 radians with respect to the position of the farmer.
If the chicken starts from the center and moves in any direction, other than towards the farmer, farmer crosses point on the circumference in that direction before chicken reaches that point.
If the chicken starts from a point on the circumference, just in time when the farmer reaches that point, he can catch the chicken if 4 times the length of he chord is equal to distance along the circumference to the other end of the chord.
In any case, it appears that the farmer must starve for the time being.