2015 AMC 10A Problem 25

Let S be a square of side length 1. Two points are chosen at random on the sides of S. The probability that the straight-line distance between the points is at least \tfrac12 is \tfrac{a-b\pi}c, where a, b, and c are positive integers with \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.