2015 AMC 12A 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 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.