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# Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Let ABCD is a quadrilateral and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively.

$$\Rightarrow$$ AS = SD, AP = BP, BQ = CQ and CR = DR.

We have to show that: PR and SQ bisect each other i.e., SO = OQ and PO = OR. Now, in $$\triangle{ADC}$$, S and R are mid-point of AD and CD.

We know that, the line segment joining the mid-points of two sides of a triangle is parallel to the third side.

Thus, by midpoint theorem,
SR || AC and SR = ($$\frac{1}{2}$$) AC ...(i)

Similarly, in $$\triangle{ABC}$$, P and Q are mid-point of AB and BC.

Thus, by midpoint theorem,
PQ || AC and PQ = ($$\frac{1}{2}$$ ) AC ...(ii)
From Eq. (i) and (ii), we get,
PQ || SR and PQ = SR = ($$\frac{1}{2}$$ ) AC

Therefore, Quadrilateral PQRS is a parallelogram whose diagonals SQ are PR.

Also, we know that diagonals of a parallelogram bisect each other. So, and bisect each other.
Hence, it is proved that PR and SQ bisect each other.