Let four points $A, B, C, D$ be concyclic.

What is the locus of points $M$ such that the circumcircle of $M A B$ and the circumcircle of $M C D$ are tangent?
This involves characterizing the points $M$ that satisfy the condition stated in the problem. Let $Z$ be the intersection point of the lines $A B$ and $C D$. The advantage of introducing this point is that it allows us to know all our radical axes. Indeed, for any point $M$, the radical axis of the circumcircle of $A B C D$ and the circumcircle of $M A B$ is the line $A B$, the radical axis of the circumcircle of $A B C D$ and the circumcircle of $M C D$ is the line $C D$, and the radical axis of the circumcircle of $M A B$ and the circumcircle of $M C D$ passes through $M$ and is concurrent with the other two radical axes (which intersect at $Z$): it is therefore the line $Z M$.

Thus, $Z$ belongs to the three radical axes, and therefore the power of $Z$ is the same with respect to the three circles. The circumcircles of $M A B$ and $M C D$ are tangent if and only if their radical axis is their common tangent at $M$, in other words, if and only if the line $Z M$ is their common tangent. This is the case if and only if the power of $Z$ with respect to the circumcircles of $M A B$ and $M C D$ is $Z M^{2}$, in other words, if and only if $Z A \times Z B = Z C \times Z D = Z M^{2}$.

In brief, knowing that the points $Z, A, B, C, D$ are fixed, the point $M$ satisfies the condition of the problem if and only if $Z M = \sqrt{Z A \times Z B}$. Therefore, the locus of our points $M$ is a circle centered at $Z$ with radius $\sqrt{Z A \times Z B}$.