2025 AIME I Problems/Problem 2

Problem

In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$ , while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$ . Suppose $AD = 4$ , $DE = 16$ , $EB = 8$ , $AF = 13$ , $FG = 52$ , and $GC = 26$ . Let $M$ be the reflection of $D$ through $F$ , and let $N$ be the reflection of $G$ through $E$ . The area of quadrilateral $DEGF$ is $288$ . Find the area of heptagon $AFNBCEM$ , as shown in the figure below.

[asy] unitsize(14); pair A = (0, 9), B = (-6, 0), C = (12, 0), D = (5A + 2B)/7, E = (2A + 5B)/7, F = (5A + 2C)/7, G = (2A + 5C)/7, M = 2F - D, N = 2E - G; filldraw(A--F--N--B--C--E--M--cycle, lightgray); draw(A--B--C--cycle); draw(D--M); draw(N--G); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(M); dot(N); label(" $A$ ", A, dir(90)); label(" $B$ ", B, dir(225)); label(" $C$ ", C, dir(315)); label(" $D$ ", D, dir(135)); label(" $E$ ", E, dir(135)); label(" $F$ ", F, dir(45)); label(" $G$ ", G, dir(45)); label(" $M$ ", M, dir(45)); label(" $N$ ", N, dir(135)); [/asy]

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2025 AIME I Problems/Problem 2

Problem