So this is a question I've pondered for a pretty long time and have decided to see if other people can come up with a solution. One of my math teachers posed a question that went something like this: "Three people stand at the vertices of an equilateral triangle of arbitrary side length s, and and a certain moment in time they all start moving towards someone opposite them (cyclically, so the person on vertex A for example moves towards the person on vertex B, the person on vertex B moves towards the person on vertex C, and the person on vertex C moves towards the person on vertex A), each with constant speed k. However, instantaneously as one starts moving the person moving towards that person has to adjust his direction so that he is now pointed towards where the guy is, after he has moved an infinitessimal amount. Find the parametric equation of the three-person system, or mathematically describe the three people as they eventually converge towards the center."

I know that the path will look like three spirals, and if at any time t you connect the points where the 3 people are, it will be another equilateral triangle but rotated and dilated down to a smaller size. However, I don't know how to solve it. My teacher gave some advice saying that using a physics property would make it much easier than using differential equations, and I know physics, but I can't think of an easy way to solve it.

On a related note, to "simplify" this I posed my own problem where a guy starts on the origin and another guy starts on the x axis a distance of a away from the origin and moves vertically (perpendicular to the x axis) at a constant speed while the guy on the origin moves with the same constant speed but always in the direction of the guy moving perpendicular and straight. This proved to be equally challenging, and I set up a differential equation but got something like dy/dx = -(d^2x/d^2y). I noted that the graph of the motion of the guy starting on the origin looks like sec(x) because it starts with dy/dx = 0 and after an infinite amount of time dy/dx = infinity, which is like when x approaches pi/2 for sec(x). However, I couldn't really express that parametrically either and so I pose these questions to all of you. I hypothesize that, in the second question, the first guy remains a distance a away from the second guy at all times.

Sounds like your teacher posed the Mice Problem. I will try my best to work on that problem, myself. All I can do right now is point you in the right direction for researching the problem. As for the problem you created, the solution is a type of pursuit curve.

Wow those are very interesting, thanks a lot. I never suspected they could be generalized that well.