Question 82
Let $ABC$ be an arbitrary triangle. A circle passes through $B$ and $C$ and intersects the lines $AB$ and $AC$ in $D$ and $E$, respectively. The projections of the points $B$ and $E$ on $CD$ are denoted by $B'$ and $E'$, respectively. The projections of the points $D$ and $C$ on $BE$ are denoted by $D'$ and $C'$, respectively.
Prove that the points $B'$, $D'$, $E'$ and $C'$ lie on the same circle.
Prove that the points $B'$, $D'$, $E'$ and $C'$ lie on the same circle.