Can two circles intersect each other at right angles, such that one circle passes through the center of the other circle? More precisely, do there exist two intersecting circles such that the tangent lines at the intersection points make an angle of 90 degrees, and one of the circles passes through the center of the other? I think the answer is no, and that this can only be the case if the circle passing through the center of the other circle is actually a line. I'm not sure how to show this though. Can someone give a proof or a counterexample?

No, this is not possible.
Here's a hint. Consider the following points:

*

*$A$: center of first circle (say radius $r_1$),

*$B$: center of second circle (say radius $r_2$),

*$C$: one point of intersection of the two circles.

Then, by the right-angle condition, you see that $ABC$ is a right-angled triangle. However, note that the sides are $r_1$, $r_2$, and $r_2$ with the right-angle being in between the sides $r_1$ and $r_2$. Check that this is not possible.
