Question

In ∆ ABC, D, E and F are mid-points of sides AB, BC and AC respectively. Prove that

AE and DF bisect each other.​

Answer 1

Answer:

Let G be the centroid of triangle ABC. Given E and F are the mid points of BC and AC respectively. Thus, by mid point theorem, AD∥EF

AB=2EF

AD=EF (I) (D is mid point of AB)

Now, In △ADG and △GEF,

∠AGD=∠EGF (Vertically opposite angles)

AD=EF (From I)

∠ADG=∠GFE (Alternate angles for parallle lines EF and AD)

△ADG≅△EGF (ASA rule)

Thus, AG=GE (Corresponding sides)

Also, DG=GF (Corresponding sides)

Thus, AE and DF bisect each other at G.

solution