Consider the following set up (you can follow on desmos):
In a plane we have to circles that intersect at 2 point, circle $A$ and circle $B$, with centers at $A$ and $B$ respectively. For simplicity assume that $A$ and $B$ are in the $Ox$ axis.
We take any point $P_0$ in the upper half-plane ($y_{P_0}>0$) and the construct the following sequence:
If n is even, $P_{n+1}$ is the point on the circle $A$ collinear with $A$ and $P_n$
If n is odd, $P_{n+1}$ is the point on the circle $B$ collinear with $B$ and $P_n$, all while in the same half plane.
It seems that the sequence converges to the intersection of the 2 circles. My question is why this sequence converges, I'm not able to prove it. I've tried solving this problem by writing the analytical equations for everything and tried to get something from there, but I got nowhere. Any book or article that covers this or something similar may suffice.
It is interesting to see that modifying the setup can have the same result. It may be the case that for any 2 curves in the upper half-plane that have an unique intersection point behave like the circles (intersect the $Ox$ axis twice) and each has corresponding point such that define the recurrence relation, the corresponding sequence converges to the intersection.
What you have discovered here is the Method of Alternating Projections. This method is for finding a point of intersection of two (typically closed, convex) sets $C$ and $D$ with non-empty intersection, and was, I am led to believe, first studied by Von Neumann.
The method is simple: as a closed, non-empty convex set $C$ has a unique nearest point, we can form the nearest point map, or metric projection $P_C$. We iterate:
\begin{align*} y_n &= P_Cx_n \\ x_{n+1} &= P_Dy_n, \end{align*}
or just $x_{n+1} = P_DP_Cx_n$, i.e. project onto one set, then project the new point onto the other.
This is precisely what you are doing here: by taking the line between your iterates and the origin of the other circle, and intersect it with the sphere, you are essentially projecting onto the convex disc.
Bregman proved that, in a Hilbert space, alternating projection iterates converge weakly to a point in the intersection (though, interestingly, not necessarily the nearest point of $C \cap D$ to the one you start from!). In finite-dimensions, this is equivalent to regular (strong) convergence, though Hundal showed that, in infinite-dimensions, strong convergence may not occur.
If you want a proof of convergence in finite dimensions, you can look here for example.