Here is the diagram that luke sent.
You just have to put together three familiar circle theorems:
- If O is the center of your circle ABNM and P is outside, then angle NPM = (1/2)(angle NOM - angle AOB).
- If O is the center of your circle ABNM and P is inside, then angle NPM = (1/2)(angle NOM + angle AOB).
- (Using neutral letters for this theorem because you will have to apply it to both your circles) if X,Y,and Z are three points on a circle whose center is C, then when X and C are on the same side of the line YZ, angle YXZ = (1/2) angle YCZ; when on opposite sides, angle YXZ + (1/2) angle YCZ = 180 degrees.