AMC 12 2025a 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 \textbf{(B)}~17 \textbf{(C)}10+4\sqrt{17} +2\sqrt{13} \textbf{(D)}\frac{113}{3} \textbf{(E)}~54