If one of PQ, PR, PS is equal to one of QR, QS, the points determine a parallelogram.

Ex:

P=(-1,0,2)

Q=(3,4,-1)

R=(3,2.-3)

S=(-1.-2,0)

PQ=(4,4,-3)

PR=(4,2,-5)

PS=(0,-2,-2)

QR=(0,-2,-2), it’s a parallelogram

QS=(-4,-6,1)

Since QR equals PS, it follows immediateley that QR and RS are adjacent sides of the parallelogram and the area is:

A=|QRXRS|

If you change S to (-1,-2,1), the conditions are not met and P,Q.R,S is not a parallelogram

This proof did not show up on an Internet search, which is why I post it here.