I was trying to maximize the distance between points on a triangle for a program the other day. Snooping around led me to the circle packing problem. However, the circle packing problem assumes that you also want to maximize distance from the borders. What if you can also place your points ON the triangle? I've been trying to figure it out by projecting onto the triangle and using calculus but that gets really nasty. Is there a geometric way to do it? Solutions for 2, 3, 4, and 6 points are pretty obvious, but I can't figure out the rest.
For example, this haphazardly drawn sketch would be the way to optimize 4.
In other words: how can we maximize the pairwise distance of points placed on or within an equilateral triangle? Can we extend this problem to other polygons?
Also, we're looking to make them both as equidistant and as maximized as possible. I'm not really certain how we could measure this, but the problem would be trivialized if we just stuck all of them into one corner and put the other one in an opposite corner. I guess the problem might be better worded as "find the distance c for which n points in or on an equilateral triangle cannot be placed without having at least one pair of points be at most c apart."