Purple Comet? Problem 26

The incircle of $\triangle ABC$ is tangent to sides $\overline{BC}, \overline{AC}$ and $\overline{AB}$ at $D,E$ , and $F$ , respectively. Point $G$ is the intersection of lines $AC$ and $DF$ as shown. The sides of $\triangle ABC$ have lengths $AB=73, BC=123$ , and $AC=120$ . Find the length $EG$ .