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The diagonals of a quadrilateral ABCD in...

The diagonals of a quadrilateral ABCD intersect each other at the point O such that `(A O)/(B O)=(C O)/(D O)`. Show that ABCD is a trapezium.

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GIVEN A quadrilateral ABCD whose diagonals AC and BD intersects at a point O such that
`(AO)/(OC)=(BO)/(OD)`

TO PROVE ABCD is a trapezium, i.e., `AB||DC`.
CONSTRUCTION Draw `EO||DC`, meeting AD at E.
PROOF In `Delta ACD, EO||DC`.
`:. (AO)/(OC)=(AE)/(ED)" "` [ by Thale's theorme]
But ,` (AO)/(OC)=(BO)/(OD)` (give)
`:. (BO)/(OD)=(AE)/(ED)rArr (DO)/(OB)=(DE)/(EA)" in" DeltaAB`.
So, `EO||AB " "` [ by the converse of Thale's theorem]
But, `EO||DC`
Hence, `AB||DC, i.e., ABCD` is a trapezium.
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