A circle with radius $6$ is centered at $(6,6)$. Line segments of length $c_i$ connect the points $(0,a_i)$ and $(b_i,0)$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers. How many such tangent segments have $c_i=5$, the minimum possible length?

1. For $r=6$ the ``small'' solutions (with $a,b<2r=12$) correspond to divisors $d$ of $2r^2=72$ strictly between $r=6$ and $2r=12$: these are $d=8,9$.
2. Each such $d$ gives one segment with $c=d+\frac{72}d-12$; both $d=8$ ($c=8+9-12=5$) and $d=9$ ($c=9+8-12=5$) give $c=5$.
3. So there are $\boxed2$ segments with $c_i=5$.