Team Selection Test

Let the lines $GE$ and $AB$ meet at a point $M$. Let $\angle GAD=\angle BAE=a$, $\angle GAB=b$. By angle chasing, we have $\angle GED=\angle MEF=\angle MAE=a$. Therefore, the points $A$, $E$, $G$, $D$ are concyclic. We also get $DGA=\angle DEA=c$. Let $\angle ABE=d$. We obtain that $\angle BEG=180^{\circ }-2a-d-c$. Using the condition given, we get $\angle GCB=2a+c+d$. Hence, $\angle BEG+\angle BCG=180^{\circ }$. Hence, we conclude that the points $B$, $E$, $G$, $C$ are concyclic. The line $BC$ is the radical axis of the circles $(ABCD)$, $(BCGE)$. The line $EG$ is the radical axis of the circles $(AEGD)$, $(BCGE)$. The line $AD$ is the radical axis of the circles $(ABCD)$ and $(AEGD)$. Therefore, these three lines should be concurrent.