How can you find the midpoint between two random points on a circle?

Hey people! The question is basically in the title. Say you have two points on your standard circle, point 1: (x,y,z) and point 2: (u,v,c). (***This is meant to be used in a 3D dimensional space.).

I want to basically be able to place these two points anywhere on said circle (anywhere except both in the same spot) and derive the center between these two points, all the while keeping this midpoint on the circle as well.

Theoretically, since the midpoint lays on the circle and isn't linear, there are two "midpoints," one closer and the other further away in comparison. Thankfully, this problem would only arise in circles whose sweep goes past 180 degrees (a semi circle). I'm only dealing with arches up to 90 degrees so I only have to worry about the short midpoint.

So graphically, there's what I'm talking about:

CGSociety Stuff :: Arch Question Diagram picture by Korinkite - Photobucket

In this picture, the two yellow dots represent the two random points on the circle that I picked and the red is the shorthand midpoint. The long midpoint that I mentioned before, the one that I can effectively negate since I only work with arches of 90 degrees (or pi/2 jeez don't correct me), is signified with a good old green X over the green dot.

So back to the main question. In this setup that I have described above, a standard circle in a 3D space, is there a formula that I can use to find the "midpoint" between two random points of the circle?