In triangle $ABC,$ $\overline{AD}$ bisects $\angle BAC,$ $\overline{AE}$ is perpendicular to $\overline{BC},$ and $F$ and $G$ are the midpoints of $\overline{AB}$ and $\overline{AC},$ respectively. Given that $AB = 16,$ $AC=13,$ $AH=13,$ and $AI=8,$ and $\angle BAC = 90^{\circ},$ which of the following triangles has an area equal to half the area of $\triangle ABC?$