Given: A parallelogram ABCD in which P and Q are the mid-points of the sides AB and CD respectively. AQ intersects DP at S and BQ interects CP respectively. AQ intersects DP at S and BQ intersects Cp at R.

To prove: PRQS is a parallelogram.

`"Proof:"" "DC"||"AB" (because""opposite sides of a parallelogram are parallel")`

`implies" "AP"||"QC`

`" "DC"||"AB" "(because"opposite sides of a parallelogram are equal")`

`implies" "1/2DC=1/2AB`

`implies" "QC=AP" "(because"P is mid-point of AB and Q is mid-points of CD")`

`implies" "APCQ"is a parallelogram."" "(becauseAP"||""QCand QC=AP)`

`therefore" "AQ"||"PC" "(because"opposite sides of a parallelogram are parallel")`

`implies" "SQ"||"PR"`

`"Similarly,"" "SP"||"QR`

`therefore` Quadrilateral PQRS is a parallelogram.