`P ,Q` and `R` are, respectively, the mid-points of sides `B C ,C A` and `A B` of a triangle `A B C` , `P R` and `B Q` meet at `XdotC R` and `P Q` meet at `Y` . Prove that `X Y=1/4B Cdot`

Explanation

Construction: Join `X` to `Y`.
To prove: `XY=1/4BC`
Proof: `R` and `Q` are the mid-points of `AB` and `AC`.
`therefore RQ` parallel to `BC` and `PQ=1/2BC` …(i) (By mid-point theorem)
`RQ` parallel to `BP` and `RQ=BP` (`∵P` is mid-point of `BC`)
`BPQR` is a parallelogram.
`∵`Diagonals is a parallelogram bisect each other.
`∵X` is the mid-point of `PR`.
Similarly, `PCQR` is a parallelogram.
`Y` is the mid-point of `PQ`
Now In `ΔPRQ`,
`X` and `Y` are mid-points of sides `PR` and `PQ` respectively.
`XY=1/2RQ` …… (ii)
From (i) and (ii), we get:
`XY=1/2RQ`
`XY=1/2 (1/2BC)=1/4BC`
`therefore XY=1/4BC`
Hence proved.