Question 9
Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing four $6$-$8$-$10$ triangles from the corners of a $23 \times 27$ rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\overline{AH},$ and partition the octagon into $7$ triangles by drawing segments $\overline{JB},$ $\overline{JC},$ $\overline{JD},$ $\overline{JE},$ $\overline{JF},$ and $\overline{JG}.$ Find the area of the convex polygon whose vertices are the centroids of these $7$ triangles.
