Question 183
Let $A$, $B$, $C$ be three points on a circle $\Gamma$, and let $L$ denote the midpoint of segment $BC$. The perpendicular bisector of $BC$ intersects the circle $\Gamma$ at two points $M$ and $N$, such that $A$ and $M$ are on different sides of line $BC$. Let $S$ denote the point where the segments $BC$ and $AM$ intersect. Line $NS$ intersects the circumcircle of $\triangle ALM$ at two points $D$ and $E$, with $D$ lying in the interior of the circle $\Gamma$.
a.
Prove that $M$ is the circumcentre of $\triangle BCD$.
b.
Prove that the circumcircles of $\triangle BCD$ and $\triangle ADN$ are tangent at $D$.
a.
Prove that $M$ is the circumcentre of $\triangle BCD$.
b.
Prove that the circumcircles of $\triangle BCD$ and $\triangle ADN$ are tangent at $D$.