2025 AMC 10A Problem 25

A point P is chosen at random inside square ABCD. The probability that AP is neither the shortest nor the longest side of \triangle APB can be written as \dfrac{a + b\pi - c\sqrt{d}}{e}, where a, b, c, d, and e are positive integers, \gcd(a, b, c, e) = 1, and d is not divisible by the square of a prime. What is a + b + c + d + e?

\textbf{(A)} 25 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 27 \qquad \textbf{(D)} 28 \qquad \textbf{(E)} 29

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