P and Q are the mid-points of the opposite sides AB and CD of parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.

Question

Philoid · Accepted Answer

Given, P and Q are mid-points of AB and CD.

Now, AB||CD,

AP||QC

Also, AB = DC

AB = DC

AP = QC

Now, AP||QC and AP = QC

APCQ is a parallelogram.

AQ||PC or SQ||PR

Again,

AB||DC AB = DC

BP = QD

Now, BP||QD and BP = QD

BPDQ is a parallelogram

So, PD||BQ or PS||QR

Thus, SQ||RP and PS||QR

PQRS is a parallelogram.