Concentric Circles do not Intersect

Theorem

If two circles are concentric, they share no points on their circumferences.

Alternatively, this can be worded:

If two concentric circles share one point on their circumferences, then they share them all (that is, they are the same circle).

Proof

This follows directly from:

Intersecting Circles have Different Centers

Touching Circles have Different Centers.

$\blacksquare$