Školjka

%V0 Let $ABC$ be an acute-angled triangle, and let $w$ be the circumcircle of triangle $ABC$. The tangent to the circle $w$ at the point $A$ meets the tangent to the circle $w$ at $C$ at the point $B^{\prime}$. The line $BB^{\prime}$ intersects the line $AC$ at $E$, and $N$ is the midpoint of the segment $BE$. Similarly, the tangent to the circle $w$ at the point $B$ meets the tangent to the circle $w$ at the point $C$ at the point $A^{\prime}$. The line $AA^{\prime}$ intersects the line $BC$ at $D$, and $M$ is the midpoint of the segment $AD$. a) Show that $\measuredangle ABM = \measuredangle BAN$. b) If $AB = 1$, determine the values of $BC$ and $AC$ for the triangles $ABC$ which maximise $\measuredangle ABM$.