2025 AMC 12A Problem 11

The orthocenter of a triangle is the concurrent intersection of the three (possibly extended) altitudes. What is the sum of the coordinates of the orthocenter of the triangle whose vertices are A(2, 31), B(8, 27), and C(18, 27)?

\textbf{(A)} 5 \qquad \textbf{(B)} 17 \qquad \textbf{(C)} 10 + 4\sqrt{17} + 2\sqrt{13} \qquad \textbf{(D)} \dfrac{113}{3} \qquad \textbf{(E)} 54

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