Question

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

Answer 1

given
AD is the median of ΔABC and E is the midpoint of AD
Through D
 draw DG || BF
In ΔADG
 E is the midpoint of AD and EF || DG
By converse of midpoint theorem we have
F is midpoint of AG and AF = FG  ..............1
Similarly, in ΔBCF 
D is the midpoint of BC and DG || BF   
G is midpoint of CF and FG = GC ..............2
From equations 1 and 2
we will get
AF = FG = GC ........3
 AF + FG + GC = AC
AF + AF + AF = AC (from eu 3)
3 AF = AC
AF = (1/3) AC