The following problem has been bothering quite a while. I guess there is a gap in my school knowledge of geometry, but I do not know how to show that:
Prove or disprove that given two circles with centers $O_1, O_2$ of the same radius $r$, which intersect at two points $A, B$, the intersection of the circles is entirely contained in the circle with the center $M$ at the midpoint of the line $O_1O_2$ and radius $AM$.
In the picture below: $M$ would be the origin, and $A$ and $B$ are two points, where the circles intersect. The desired circle that is supposed to contain the intersection is drawn in green.