Open Contests

The radius $AO$ of the circle $c$ is perpendicular to the common tangent to circles $c$ and $c_1$ at the point $A$, hence $AO$ is a diameter of the circle $c_1$. Similarly $BO$ is a diameter of the circle $c_2$.

If the circles $c_1$ and $c_2$ are tangent at the point $O$ (Fig. 1), then the diameters $AO$ and $BO$ are both perpendicular to the common tangent to $c_1$ and $c_2$ at the point $O$, whence the lines $AO$ and $BO$ coincide, i.e. $O$ lies in the segment $AB$.

If the circles $c_1$ and $c_2$ intersect at $O$ (Fig. 2), then let $M$ be the other intersection point of the circles. Since $\angle AMO=90^{\circ }$ and $\angle BMO=90^{\circ }$ (angles at the circumference supported by a diameter), the lines $AM$ and $BM$ coincide and $M$ lies in the segment $AB$.

Figure 1

Figure 2