Shortest polygonal line that connects three disjointing circles

Question

Given three disjointing circles, how to find the shortest polygonal line (consisting of two line segments) that connects the three circles (a line segment connects circle A and B if it starts with some point on A and ends with some point on B)

Answer 1

The shortest polygonal path $\gamma=\gamma_{ABC}$ from disk $A$ to disk $B$ to disk $C$ is found as follows: If the segment $[m_A,m_C]$ connecting the midpoints of $A$ and $C$ hits the disk $B$ then $\gamma$ is part of this segment. Otherwise $\gamma$ is lying on two lines $\ell_A$ and $\ell_C$ through $m_A$ and $m_C$. These lines intersect at a point $P$ on the circle $\partial B$, whereby the following condition is satisfied:

$(*)$ The line $\ell_B:=m_B\vee P$ is an angle bisector of the lines $\ell_A$, $\ell_C$ intersecting at $P$.

Reason for $(*)$: The shortest path from $m_A$ to $B$ to $m_C$ (that's what we need) is found as follows: Find the ellipse $\epsilon$ with foci $m_A$ and $m_C$ that touches $B$ at a point $P\in\partial B$. Then $\epsilon$ and $\partial B$ have a common tangent at $P$, and the rays $[m_A,P]$, $[P,m_C]$ are reflected on this tangent.

Unfortunately this leads to complicated equations for the angles involved.

One would have to solve the above problem for all three tours $A\to B\to C$, $\ B\to C\to A$, $C\to A\to B$ separately and then take the shortest of the three tours so obtained.