2011 AMC 12A Problem 13

Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N.$ What is the perimeter of $\triangle AMN?$

$\textbf{(A)}\ 27 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 33 \qquad\textbf{(D)}\ 36 \qquad\textbf{(E)}\ 42$

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