Given
a circular arc described by three points: $A_1$, $A_2$ and $A_3$
a line segment described by two points: $B_1$ and $B_2$
how could I determine the maximum distance between any point on the arc and its corresponding closest point on the line segment?
Construct the midline perpendicular to $A_1A_2$ as well that of $A_2A_3$. These will intersect in $M$. Then construct the perpendicular from $M$ wrt. the line $B_1B_2$. Finally you will have to take on that very line just the same distance as between $A_1$ and $M$. This will determine that farthest point $C$ of the arc. Now you just have to consider the distance between that point $C$ and the line $B_1B_2$.
--- rk