ABCD is a rectangle and P, Q, R, and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

In ΔABC, P and Q are the mid-points of sides AB and BC. Using Mid-point Theorem, PQ∥AC and PQ=1/2AC.. (1)
Similarly, In ΔADC, R and S are the mid-points of sides CD and AD. Using Mid-point Theorem SR∥AC and SR=1/2AC.. (2)
From (1) and (2), we get PQ∥SR and PQ=SR=1/2AC. (3)
similarly, PS∥QR and PS=QR=1/2BD (4)
Also, AC=BD [diagonals of a rectangle are equal]
⇒1/2AC=1/2BD
⇒PQ=SR=PS=QR
from (3) and (4)
∴PQRS is a Rhombus