For the zigzag part, just think about it, if the chicken runs in the opposite direction of the human and start zigzagging, it will be closer and closer to the fence, but the human will only be going back and forth around its initial point. Assuming the human will always choose direction that has the shortest distance toward the straight line that the chicken is heading.
To show that it is impossible to beat the human without changing direction:
Let Vh be the tangential speed of human
Let Vc be the speed of chicken
Let Tc be the time for chicken to reach the fence (opposite of human)
Let Th be the time for human to reach that same point on the fence
Let w be the angular velocity of human
Let R be the radius of the circle
We have:
Vh = 4*Vc, so R*w = 4*Vc (since Vh = R*w)
Time for human to reach the end point:
theta(final) = theta(inital) + w*Th
pi = 0 + w*Th (since initial angle is 0 and final angle is pi)
Th = pi/w
Plug in for w = 4*Vc/R:
Th = pi*R/(4*Vc)
Time for chicken to reach the end point:
Tc = R/Vc
Therefore, Th/Tc = pi/4
Sorry this is really messy...