Question 83
A triangle $ABC$ and a point $D$ on the line segment $AC$ are given. Let $M$ be the midpoint of $CD$ and let $\Omega$ be the circle through $B$ and $D$ tangent to $AB$. Let $E$ be the point such that $\triangle MDB \sim \triangle MBE$ and such that $D$ and $E$ lie on opposite sides of the line $MB$.
Show that $E$ lies on $\Omega$ if and only if $\angle ABD = \angle MBC$.
Show that $E$ lies on $\Omega$ if and only if $\angle ABD = \angle MBC$.