`P` is the mid-point of side `A B` of a parallelogram `A B C D` . A Line through `B` parallel to `P D` meets `D C` at `Q` and `A D` produced at `R` . Prove that (i) `A R=2B C` (ii) `B R=2B Qdot`

Explanation

(i) In `/_\ARB`, `P` is the mid-point of `AB` and `PD` parallel to `BR`.
`:. D` is a mid-point of `AR` [Converse mid point theorem]
`:. AR=2AD`
But `BC=AD` [Opposite sides of parallelogram are equal]
`:. AR=2BC`

(ii) `ABCD` is a parallelogram.
`:. DC` parallel to `AB` and `DQ` parallel to `AB`
Now, In `/_\ARB`,
`D` is a mid-point of `AR` and `DC` parallel to `AB`
`therefore Q` is a mid point of `BR` [Converse mid-point theorem ]
`BR=2BQ`