ABC is a triangle. AD is a median of ABC. E is the midpoint of AD and produced to meet AC at F. prove that AF=1/3AC
Given: AD is the median of ΔABC. E is the mid point of AD. BE produced meets AD at F
To Prove :
Construction: From Point D, draw DG BF.
Proof: In ΔADG, E is the mid-point of AD and EF||DG.
∴F is the mid point of AG [Converse of the mid point theorem]
⇒ AF = FG ... (i)
In ΔBCF, D is the mid point of BC and DG||BF
∴ G is the mid point of CF
⇒ FG = GC ... (ii)
From (i) and (ii), we get,
AF = FG = GC ... (iii)
Now, AF + FG + GC = AC
⇒ AF + AF + AF = AC [Using (iii)]
⇒ 3AF = AC