1. Define the circles and points:
Let $k$ and $k'$ be two concentric circles centered at $O$, with $k'$ being larger than $k$. A line through $O$ intersects $k$ at $A$ and $k'$ at $B$ such that $O$ separates $A$ and $B$. Another line through $O$ intersects $k$ at $E$ and $k'$ at $F$ such that $E$ separates $O$ and $F$.

2. **Define the circles with diameters $AB$ and $EF$:**
Let $m$ be the circle with diameter $AB$ and $n$ be the circle with diameter $EF$.

3. Draw tangents to the circles:
Draw tangents to $m$ at $A$ and $B$. Let these tangents meet at point $M$. Similarly, draw tangents to $n$ at $E$ and $F$. Let these tangents meet at point $N$.

4. **Radical axis of circles $m$ and $n$:**
The points $M$ and $N$ have equal powers with respect to $m$ and $n$. Therefore, the line $MN$ is the radical axis of $m$ and $n$.

5. **Intersection of the circumcircle of $\triangle OAE$ with $MN$:**
Observe that $M$ lies on the circumcircle of $\triangle OAE$. However, this is not the required point since its power with respect to $n$ is $MF^2 > 0$. Let $P$ be the projection of $O$ on $MN$. This $P$ is the second intersection of the circumcircle of $\triangle OAE$ with $MN$.

6. Angles and cyclic quadrilaterals:
We claim that $P$ is the required point. Note that $\angle OPE = \angle EAO$ since $A, O, E, P, M$ are concyclic. Since $OA = OE$, we have $\angle OEA = \angle OAE = \angle OPE$.

7. **Cyclic quadrilateral $PONF$:**
Also, $\angle FPN = \angle FON = \angle NOB$ since $PONF$ is cyclic and $FN = NB$. Thus, $\angle EOB = 2\angle OPE = 2\angle FPN$.

8. Conclusion:
Therefore, $\angle FPE = \angle EPN + \angle FPN = \angle EPN + \angle OPE = \frac{\pi}{2}$. So, $P \in n$ and since $MN$ is the radical axis, $P$ is also on $m$.

$\blacksquare$