Suppose we are given a ray $\rho_a$ beginning at point $a$ and a ray $\rho_b$ beginning at point $b$. I want to find a circle $C_a$ tangent to $\rho_a$ at point $a$ and another circle $C_b$ tangent to $\rho_b$ at point $b$ such that there exists some point $c$ and a line $\ell$ passing through $c$ such that $C_a$ and $C_b$ intersect at $c$ tangent to $\ell$.
Note that if we insisted on a particular point $c$ and line $\ell$ given ahead of time, then each circle would be over-constrained. However, since we don't insist on a particular point $c$, I think that this should be possible.
Edit: Here are two possible solutions (approximate, b/c it's hard to draw)