AD is the median of the triangle ABC and E is the midpoint AD, BE produced meets AC in F. Prove that AF=1/3 AC?

We have

given: ABC is a triangle . AD is the median of the triangle. and

E is the mid-point of AD. BE produced meets AC at F.

To prove: AF=1/3 AC

construction: draw a line DG parallel to BF, intersecting AC at G.

Proof: 

in the triangle ADG,

E is the mid-point of AD. and .

by the mid-point theorem, 

F is the mid-point of AG.

AF=FG............(1)

in the triangle CFB,

D is the mid-point of BC. and .

by the mid-point theorem, 

G is the mid-point of CF.

CG=FG............(2)

by (1) and (2), AF=FG=CG=AC/3

which is the required result.