Question 96
Let $D$ be an interior point on the side $BC$ of an acute-angled triangle $ABC$. Let the circumcircle of triangle $ADB$ intersect the side $AC$ again at $E$ ($\neq A$) and the circumcircle of triangle $ADC$ intersect the side $AB$ again at $F$ ($\neq A$). Let $AD$, $BE$ and $CF$ intersect the circumcircle of triangle $ABC$ again at $D_1$ ($\neq A$), $E_1$ ($\neq B$) and $F_1$ ($\neq C$), respectively. Let $I$ and $I_1$ be the incenters of triangles $DEF$ and $D_1E_1F_1$, respectively. Prove that $E$, $F$, $I$, $I_1$ are concyclic.