its been a while since i took trig. so my terminology may be incorrect, but if the given coordinates are on the circumference of the circle then there could be two more sets of coordinates on the circle that are 120 degrees apart. one set clockwise 120 degrees and another set counter clockwise 120 degrees. the center of the circle would be at (0,0) i believe.
You posted this in a trigonometry forum. The answer I will give may not look correct in trig terms.
This is done with a rotation matrix, a transformation.
is the resulting point of rotating the point about through an angle of .