Given: ABCD is a rectangle and P, Q, R, S are their midpoints

To Prove: PQRS is a rhombus

Proof: In ABC,

P and Q are the mid points

So, PQ is parallel AC

And,

PQ = AC (The line segment joining the mid points of 2 sides of the triangle is parallel to the third side and half of the third side)

Similarly,

RS is parallel AC

And,

RS = AC

Hence, both PQ and RS are parallel to AC and equal to AC.

Hence, PQRS is a parallelogram

In triangles APS & BPQ,

AP = BP (P is the mid-point of side AB)

∠PAS = ∠PBQ (90^o each)

AS = BQ (S and Q are the mid points of AD and BC respectively and since opposite sides of a rectangle are equal, so their halves will also be equal)

APS BPQ (By SAS congruence rule)

PS=PQ (By c.p.c.t.)

PQRS is a parallelogram in which adjacent sides are equal.

Hence, PQRS is a rhombus.