According to the question,
We have,
P is the mid-point of the sides AB
Q is the mid-point of the sides BC
R is the mid-point of the sides CD
S is the mid-point of the sides DA
Also,
AC ⊥ BD
∠COD = ∠AOD = ∠AOB = ∠COB = 90o
In ΔADC, by mid-point theorem,
SR = ½ AC
And, SR||AC
In ΔABC, by mid-point theorem,
PQ = ½ AC
And, PQ||AC
So, we have,
PQ||SR and SR = PQ = ½ AC
Similarly,
SP||RQ and SP = RQ = ½ BD
Now, in quadrilateral EOFR,
OE||FR and OF||ER
So, we get,
∠EOF = ∠ERF = 90o
Hence, PQRS is a rectangle.