AD is the median of a triangle ABC. E is mid-point of AD. BE produced meets AC at F. Show that AF=1/3AC

Construction = Through D, draw DG parallel to BF

 

 

In triangle ADG,
  E is the midpoint of AD and EF parallel DG
  According to the Reverse of Midpoint theorem, AF = FG-- ( i )
  In triangle BCF, 
  D is the midpoint of BC and DG parallel to BF   
  According to Reverse of Midpoint theorem, FG = GC -- ( ii )
  From equation ( i ) & ( ii ),
  AF = FG= GC 
  Now, AF + FG + GC = AC
  AF + AF+ AF = AC 
  3 AF = AC
  AF = 1/3 AC