In ` A B C ,A D` is the median through `A` and `E` is the mid-point of `A D`. `B E` produced meets `A C` in `F` (Figure). Prove that `A F=1/3A Cdot`

Explanation

Given In a `Delta`ABC, AD is a median and E is the mid-point of AD.
Construction Draw DP||EF.
Proof In `Delta`ADP, E is the mid-point of AD and EF||DP.
So, F is mid-point of AP. `" "` [by converse of mid-point theorem]

In `Delta`FBC, D is mid-point of BC and DP||BF.
So, P is mid-point of FC.
Thus, `" "AF=FP=PC`
`therefore" "AF=(1)/(3)AC" "` Hence proved.