8.4 The Mid-Point Theorem (MPT)

Given: Quad $ABCD$. $P, Q, R, S$ are mid-points of $AB, BC, CD, DA$ respectively.

Construction: Join diagonal $AC$.

In $\Delta ABC$: $P$ and $Q$ are mid-points. By Mid-Point Theorem, $PQ || AC$ and $PQ = \frac{1}{2} AC$.

In $\Delta ADC$: $S$ and $R$ are mid-points. By Mid-Point Theorem, $SR || AC$ and $SR = \frac{1}{2} AC$.

From above steps: $PQ || SR$ (since both parallel to $AC$) and $PQ = SR$ (both half of $AC$).

Since one pair of opposite sides ($PQ, SR$) is equal and parallel, $PQRS$ is a parallelogram.