Question 57
Generalization: Given a segment $AB$ and a point $M$ inside of it, construct circle $\omega_{l}$ centered at $O_{l}$ passing through $A$ and $M$ and $\omega_{r}$ centered at $O_{r}$ passing through $M$ and $B$ so that $O_{l}$ and $O_{r}$ are on the same side of $AB$ and $\angle A O_{l} M = \angle M O_{r} B = 2x$. Then $\omega_{l}$ and $\omega_{r}$ intersect at $M$ and another point $N$. Extend $AN$ until it intersects $\omega_{r}$ again at a point $D$. Prove that $\angle DBA = x$, and moreover, all lines $NM$ pass through the same point $K$ in the plane. (Note that for triangles we have $x = 60^{\circ}$, and for squares we have $x = 45^{\circ}$.)