KöMaL 2011 B4406

Lines $e_1$ and $e_2$ are perpendicular, and $P$ is a point on one of their angle bisectors, different from the intersection. Lines $f$ and $g$ pass through $P$, and intersect the lines $e_i$ at points $F_i$ and $G_i$, respectively. Determine the locus of the intersection of the lines $F_1 G_2$ and $F_2 G_1$.