Click here to see the problem.
Eventually they all meet in the center of the square. Why?
From the looks of the gsp sketch given and the one I created several times, I found that that the four dogs would eventually meet in the center of the circle because they are all running around the path of a semicircle and therefore the radius of that semicircle would be the length of the side of the square created in the run. At the rate they are going they will meet in the center of the square.
Look and see:
What are the paths followed by the four dogs?
The paths the dogs are following are along semicircles.
What is the length of the path followed by each dog?
The length of the path followed by
each dog is: 
How many times does the path of each of the dogs cross a diagonal of the square? at what angle?
The path of each of the dogs crosses a diagonal of the square twice at a 45 degree angle and again at a 90 degree angle.
Suggestion: Implement the animation on a GSP sketch. Vary the speed of the animation to obtain different configurations. Click here to see the GSP Sketch (and make sure to animate by double clicking) created by Dr. Wilson or you may look at the GSS and create your own by using the one created by Shae and I by clicking here.