In a plane geometry class we were given a list of exercises that includes the following:
If a circle $L=(A,r)$ cuts the circle $K=(O,k)$ and $k^2+r^2=|OA|^2$, show that the inverse point of $A$ with respect to $K$ is the midpoint of the common chord of $K$ and $L$.
I have been stuck for a while trying to come up with a solution. I don't even know how to start. Any help is appreciated.
Hint: The "interesting" part of the problem is the condition that $k^2+r^2=|OA|^2$. This implies that the intersection points $P$ and $Q$ of the two circles satisfy $OP^2+PA^2=OA^2$. See if you can use this to construct a circle through $O$, and then show that $A$ inverts to the midpoint of $PQ$.
Translational note: I think "rope" (cuerda, presumably) should translate to "chord" (and "middle point" should be "midpoint"), but it's certainly understandable the way you wrote it.