Question 11
Let $\triangle ABC$ be a right-angled triangle with $\angle C = 90^\circ$. $CD$ is the altitude from $C$ to $AB$, with $D$ on $AB$. $\omega$ is the circumcircle of $\triangle BCD$. $\omega_1$ is a circle situated in $\triangle ACD$, which is tangent to the segments $AD$ and $AC$ at $M$ and $N$ respectively, and is also tangent to circle $\omega$.
(i) Show that $BD \cdot CN + BC \cdot DM = CD \cdot BM$.
(ii) Show that $BM = BC$.
(i) Show that $BD \cdot CN + BC \cdot DM = CD \cdot BM$.
(ii) Show that $BM = BC$.