Question 6
Let $\triangle ABC$ a triangle with $90^\circ \neq \hat{A} \neq 135^\circ$. Let $D$ and $E$ be external points to the triangle such that $DAB$ and $EAC$ are isosceles triangles with right angles in $D$ and $E$, respectively. Let $F = BE \cap CD$, and $M, N$ the midpoints of $BC, DE$, respectively.
Prove that, if three of the points $A, F, M, N$ are collinear, then the four points are collinear.
Prove that, if three of the points $A, F, M, N$ are collinear, then the four points are collinear.