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If two circles are separated is there a way to calculate the most acute angle I can move one circle without overlapping the other?

If I have two circles touching tangentially and wish to move one without overlapping the other, then they would have to move on the single tangent correct? I’m having trouble proving this.

Also, if the two circles are separated. If I know the radius of both circles and the distance between the two; is there a way to calculate the most acute angle I can move one circle without overlapping the other?

If so, what would be more useful to me is to know the distance between the two circles after I moved the one circle along a straight line on the most acute angle possible without ever overlapping each other. To clarify, the movement of the one circle would end when it was 90 degrees to the bisecting centroid of the other circle. (Not sure I expressed that correctly. Basically if I drew a bisecting line between the two circles and then another line 90 degrees to the first line bisecting the stationary circle, the distance moved would be determined when the moving circle’s center would reach the other line)

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