TITLE: Locus of intersection point of two lines
QUESTION [6 upvotes]: Consider a triangle ABC, M the midpoint of BC and a line rotating around M, intersecting AB and AC at points K and L respectively. We then draw the lines BL and CK and their intersection point is N. Find the locus of point N.

I have noticed that the locus is a line from A and parallel to the side BC but I don't know how to prove it. 
(Geometry is not my strong suit and I can recall only taking two college classes in the subject).
Any help?

REPLY [2 votes]: A more geometrical answer
Let the line through $M$ intersect $AB$ extended at $K$. Let the line $KC$ intersect the line through $A$ which is parallel to $BC$ at $N'$. (We will prove $N=N'$.)
Apply a shear, fixing $A$ and $N'$ and making $AK=KN'$. After the shear,  $KML$ is the line of symmetry of triangle $AKN'$ and so $N=N'$. Therefore this was true before the shear!