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**HallsofIvy** I think it is clear that you don't have enough information. Certainly the **rate** at which each point moves around it circle has a lot to do with this, as well as the radii of the two circles and possibly the starting points. For example, if you were to start with the two points at the points of the circles closest together, the two circles having the same radii, and the points move around the circle at the same rate, one clockwise and the other counter-clockwise, the center point will move on a line segment on the perpendicular bisector of the line segment from the center of one circle to the other. And if you have the same starting position, two circles with the same radii, and the two points moving with the same rate, both clockwise, then the locus of the midpoints will be a single point- the midpoint of the line segment between centers.