To prove LM = LN

Draw LS perpendicular to line MN

The lines BM, LS and CN being the same perpendiculars, on line MN are parallel to each other.

According to intercept theorem,

If there are three or more parallel lines and the intercepts made by them on a transversal or equal. Then the corresponding intercepts on any other transversal are also equal.

In the drawn figure, MB and LS and NC are three parallel lines and the two transversal line are MN and BC

We have, BL = LC (As L is the given midpoint of BC)