*Let *M *and* N *be the midpoints of the sides *BC *and *AD respectively in a quadrilateral ABCD *and also let *P *and* Q *be the midpoints of its diagonals *AC *and *BD *respectively. Prove that *MN *bisects *PQ*.*

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In triangle ABD , N is mid point of AD and Q is mid point of BD.

Thus by mid point theorem , QN is parallel to AB. (i)

In triangle ABC , P is mid point of AC and M is mid point of BC.

Thus by mid point theorem , PM is parallel to AB. (ii)

Thus from (i) and (ii)

NQ is parallel to PM (iii)

In triangle ADC , N is mid point of AD and P is mid point of AC.

Thus by mid point theorem , NP is parallel to DC. (iv)

and

In triangle BDC , M is mid point of BC and Q is mid point of BD.

Thus by mid point theorem , MQ is parallel to DC. (v)

from (iv) and (v)

NP is parallel to QM (vi)

From (iii) and (vi) , we have NQMP is parallelogram.

In Parallelogram ,diagonal bisect each other ,therefore MN will bisect PQ.

QED.

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