Question 42
Let $ABC$ be a triangle and let $M$ be the midpoint of the side $BC$. The circle of radius $MA$ centred at $M$ meets the lines $AB$ and $AC$ again at $B'$ and $C'$, respectively, and the tangents to this circle at $B'$ and $C'$ meet at $D$. Show that the perpendicular bisector of the segment $BC$ bisects the segment $AD$.