Question 22
Let $ABC$ be an acute triangle with the orthocentre $H$. Let $D$ be such a point that the quadrilateral $AHCD$ is a parallelogram. Let $p$ be a line perpendicular to the line $AB$ passing through the midpoint $A_1$ of the segment $BC$. Let us denote the intersection of $p$ and $AB$ by $E$ and the midpoint of the segment $A_1E$ by $F$. We denote the point in which the line parallel with the line $BD$ through $A$ intersects $p$ with $G$.
Prove that the quadrilateral $AFA_1C$ is cyclic if and only if the line $BF$ passes through the midpoint of the segment $CG$.
Prove that the quadrilateral $AFA_1C$ is cyclic if and only if the line $BF$ passes through the midpoint of the segment $CG$.