Given ` A B C`, lines are drawn through `A , B and C` parallel respectively to the sides `B C , C A and A B , ` forming ` P Q Rdot`(Figure). Show that `B C=1/2Q R`

Explanation

We know that `AR` parallel `BC` and `AB` parallel `RC`.
From the figure we know that `ABCR` is a parallelogram, so we get
`AR=BC`.... (1)
We know that `AQ` is parallel to `BC` and `QB` is parallel to `AC`
From the figure we know that `AQBC` is a parallelogram, so we get
`AQ=BC`.......(2)
By adding both the equations
`AR+QA=BC+BC`
We know that `AR+QA=QR`
So we get,
`QR=2BC`
`BC=(QR)/2`
`BC=1/2QR`
`therefore` It is proved that `BC=1/2QR`.
Hence Proved.