Both Cynic and Keith may be right, but I would like to see the math fully worked out so that we can be 100% certain. I think in both cases they will have swum into the area between the two inner circles in the answer below, which then allows them to get out safely.
The answer is he can get out safely. This is one proof, but as I said, zig zag methods may also prove successful.
Looking at the three circles in the diagram, the outer circle represents the lake. It has radius R metres. Lets assume you can swim at speed V metres per second.
We know that if you were at the centre of the lake and swam directly to the side, the cyclops could always get there before you, because it would take you R/V seconds to get to the shore and the maximum time he needs to run around half the circumference of the lake is πR/4V seconds. Since π is 3.14 approx, then π/4 is less than 1, hence the cyclops will always get there first.
In the time the cyclops takes to run a half circumference, πR/4V seconds, you can swim πR/4 metres. This is 0.79R and is represented by the distance D in the diagram (not to scale). So it is clear if you can manage to to be anywhere less than D metres from the shore when the cyclops is at the exact opposite side of the lake (180 degrees removed), you can get to the shore before he can get to your point of exit. The innermost circle of radius A represents all points where
you would NOT make it to the shore before the cyclops if he were to start 180 degrees removed from you. A is obviously R-D, or 0.21R. So if you are anywhere outside that innermost circle and the cyclops is 180 degrees removed from you, you can get to the shore before he gets to you. So how do you manage to do this?
The time taken for the cyclops to fully complete a circle is 2πR/4V or πR/2V seconds. The circle with radius B represent the circle that you can fully swim around in the same time it takes the cyclops to run around the lake. The time you take to swim around that circle is 2πB/V seconds and if this is the same time as it takes the cyclops to run around the lake, we get:
2πB/V = πR/2V or B = R/4 or 0.25R
So if you are on that circle with radius B, you can swim around it in the same time it takes the cyclops to run around the lake. Obviously if you are on a smaller circle (one with radius less than B) you can swim around that smaller circle faster than the cyclops can run around the lake.
Although I have drawn it that way, it wasn’t known until we worked the above out that the circle with radius A (where A = 0.21R) is inside the circle with radius B (where B is 0.25R). This is fortunate as it means that if we swim to anywhere in the area between the two inner circles, because we are within the circle with radius B we can swim in a circular fashion around the lake faster than the cyclops can run around it, so we will be able to get to a point where he is 180 degrees removed from where we are, but because we are outside the circle with radius A, we can then get to the shore before he can get around to the point we exit.
So to escape, you just need swim to just outside the A circle and then swim on an imaginary circle around the centre until we are 180 degrees removed from where the cyclops is. At that point we then swim directly to the shore and will be able to get out before he makes back around.