Problem: Three people want to be all gathered at the same place, and they want it to happen as soon as possible. Where should they head to?
P.S. Assume they all travel with the same speed. Think of the people as points in a plane. A person can stop and wait.
Remarks:
One possible approach is for the person in the middle to remain still and the others to head towards he/she. This is simple, but is not optimal in the general case.
If they are all on a straight line or on a circle the problem should be easier, but it's still not the general case and I don't know if solving for these special cases will shed light on the general one.
Note: I invented this problem by myself, but maybe it was already discussed/solved somewhere, if this is the case please point some reference(s).
Hint: You are trying to find a point which minimises the maximum distance to the given points / people. In general, this would be the centre of the smallest "covering circle" for the points.
In the case of three points, this would then be either the circumcentre or the midpoint of the longest side...
For more, you may want to search for "minimax facility location" problem.