Question 27
In the square $ABCD$ the points $E$ and $F$ lie on the sides $(AD)$ and $(DC)$, respectively. The diagonal $AC$ intersects $BE$ and $BF$ at the points $H$ and $G$, respectively. If $m(\angle EBF) = 45^{\circ}$ and $EG \cap HF = \{O\}$, prove that the lines $BO$ and $EF$ are perpendicular.