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.