Data : P, Q, R and S are mid-points of AB, BC, CD and DA respectively in quadrilateral ABCD.
To Prove: PR and SQ line segments bisect mutually.
Construction: Join the diagonal AC.
Proof: In ∆ADC, S and R are the mid-points of AD and DC.
∴ SR || AC
SR = \(\frac{1}{2}\)AC ………… (i)
(Mid-point Theorem)
Similarly, in ∆ABC,
PQ || AC
PQ = \(\frac{1}{2}\)AC …………. (ii)
From (i) and (ii),
SR = PQ and
SR || PQ
∴ PQRS is a parallelogram. PR and SQ are the diagonals of parallelogram PQRS.
∴PR and SQ meet at O’.