Proving that two segments have same length.

Question

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$, with $L$ situated on the side $BC$ and $K$ situated on the side $AC$. The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN || MK$. Prove that $LN = NA$.

Answer 1

Hint: Use the property: "the intersection point of the angle bisector of an angle of a triangle and the perpendicular bisector of the opposite side belongs to the circumcircle of the given triangle". Hence, quadrilateral $ABMK$ is cyclic.