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Prove that the line segments joints join...

Prove that the line segments joints joining the mid-points of the adjacent sides of a quadrilateral from a parallelogram.

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GIVEN A quadrilateral ABCD in which P,Q,R,S are the midpoint of AB,BC, CD and DA respectively.
TO PROVE PQRS is a parallelogram .
CONSTRUCTION Join AC.

PROOF In `Delta ABC, P and Q` are the midpoints of AB and BC respectively.
`:. PQ||AC " " ...(i)` [ by midpoint theorem]
In `Delta DAC, S and R` are the midpoints of AD and CD respecrtively.
`:. PQ||AC " "...(ii)` [ by midpoint theorem]
From (i) and (ii), we get `PQ||QR`.
Similarly by joining BD, we can prove that `PS||QR`.
Here, PQRS i a parallelogram.
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