Question

In ΔABC, D, E and F are the midpoints of AB, BC and AC.
If AE and DF intersect at G, and M and N are the midpoints of GB and GC respectively, prove that DMNF is a parallelogram.

Answer 1

Consider ΔABC and ΔGBC, by mid-point theorem,
2DF = BC and 2MN = BC
⇒ DF = MN      ....(i)
Consider ΔABG and ΔACG, by mid-point theorem,
2DM = AG and 2FN = AG
⇒ DM = FN     ....(ii)
From (i) and (ii), it is clear that DMNF is a parallelogram.