Question 51
In a convex pentagon $A B C D E$, let $F$ be a point on $A C$ such that $\angle F B C = 90^{\circ}$. Suppose triangles $A B F$, $A C D$ and $A D E$ are similar isosceles triangles with
$$
\angle F A B = \angle F B A = \angle D A C = \angle D C A = \angle E A D = \angle E D A. \tag{1}
$$
Let $M$ be the midpoint of $C F$. Point $X$ is chosen such that $A M X E$ is a parallelogram. Show that $B D$, $E M$ and $F X$ are concurrent.
$$
\angle F A B = \angle F B A = \angle D A C = \angle D C A = \angle E A D = \angle E D A. \tag{1}
$$
Let $M$ be the midpoint of $C F$. Point $X$ is chosen such that $A M X E$ is a parallelogram. Show that $B D$, $E M$ and $F X$ are concurrent.