$P, Q, R$ and $S$ are respectively the mid-points of the sides $A B, B C, C D$ and $D A$ of a quadrilateral $A B C D$ such that $A C ⊥ B D$ . Prove that $P Q R S$ is a rectangle.

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Solution

In $Δ$ ABC
P is midpoint of AB and Q is midpoint of BC.
By midpoint theorem,
$P Q = \frac{1}{2} A C$
Similarly in $Δ$ ADC
$R S = \frac{1}{2} A C$
$P Q = R S$
Similarly in $Δ D A B$
$P S = \frac{1}{2} B D$
Similarly in $Δ D C B$
$Q R = \frac{1}{2} B D$
$P S = Q R$
PQRS is rectangle with $P S = Q R$ and $P Q = R S$ .

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