In a triangle ABC, D, E and F are the mid-points of the sides BD, CA and AB, respectively. BE and DF intersect at
X. DE and CF intersect at Y. Find XY = ?

In ΔABC, F is the midpoint of AB and E is the midpoint of AC. ∴ By Midpoint Theorem, EF ∥ BC ∴ EF ∥ BD ----(1) ⇒ EF = BC/2 ----(2) Since D is the midpoint of BC, ⇒ EF = BD ----(3) From equation 1 and 3, ⇒ BDEF is a Parallelogram. BE and DF intersect at X. Similarly, DCEF is a Parallelogram. DE and CF intersect at Y. ∵ X and Y are the midpoints of sides DF and DE, respectively. In ΔDEF, X is the midpoint of DF and Y is the midpoint of DE. ∴ By Midpoint Theorem, ⇒ XY = EF/2 ----(4) From equation 3 and 4 ⇒ XY = BC/4