AMC 12 2015a Problem 23

Let $S$ be a square of side length 1. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\frac12$ is $\frac{a-b\pi}{c}$, where $a,b,$ and $c$ are positive integers and $\text{gcd}(a,b,c) = 1$. What is $a+b+c$?

\textbf{(A)}\ 59 \textbf{(B)}\ 60 \textbf{(C)}\ 61 \textbf{(D)}\ 62 \textbf{(E)}\ 63