show that the line segments joining the mid-points of the opposite sides of a quad bisect each other

let ABCD be the quadrilateral.

P,Q,R,S  be the mid ptsof AB,BC,CD,DA.

join diagonal AC.

Join S,R and P,Q.

in triangles  DAC,

S and R  are mid pts of sides, so,

SR II AC...   and SR=1/2 (AC)    ...Mid pt thm

similarly,

PQ II AC and PQ=1/2?(AC)

So,

PQRS  is a parallelogram as one pair of opp sides is parallel an equal.(this can be proved even by taking the other diagonal and using the fact that when both pairs of opp sides are paralle, the quad is a parallelogram)

so,

PR and SQ become the diagonals of the parallelogram PQRS, SO they must bisect each other.