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If the diagonals of a quadrilateral bise...

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

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(i) Let `bar(a),bar(b),bar(c)andbar(d)` be respectively the position vectors of the vertices A,B,C and D of the parallellogram ABCD
Then AB=DC and side `AB||` side DC.

`:.bar(AB)=bar(DC)`
`:.bar(b)-bar(a)=bar(c)-bar(d)`
`:.bar(a)+bar(c)=bar(b)+bar(d)`
`:.(bar(a)+bar(c))/(2)=(bar(b)+bar(d))/(2)` . . .(1)
The position vectors of the midpoints of diagonals AC and BD are `(bar(a)+bar(c))//2 and (bar(b)+bar(d))//2`. By (1) they are equal.
`:.` the mdpoints of the diagonals AC and BD are the same
This shows that the diagonals AC and BD bisect each other.
(ii) Conversely, suppose that the diagonals AC and BD of `square ABCD` bisect other.
i.e., they have the same midpoint.
`:.` the position vectors of these midpoints are equal.
`:.(bar(a)+bar(c))/(2)=(bar(b)+bar(d))/(2)`
`:.bar(a)+bar(c)=bar(b)+bar(d)`
`:.bar(b)-bar(a)=bar(c)-bar(d)`
`:.bar(B)=bar(DC)`
`:.bar(AB)||bar(DC)and|bar(AB)|=|bar(DC)|`
`:.` side `AB||` sides DC and AB=DC.
`:. square ABCD` is a parallelogram.
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