The diagonals of a quadrilateral ABCD intersect each other at the point O such that AOBO=CODO. Show that ABCD is a trapezium.
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Solution
Given: The diagonals of a quadrilateral ABCD intersect each other at the point O such that AOBO=CODO i.e., AOCO=BODO To Prove: ABCD is a trapezium Construction: Draw OE∥DC such that E lies on BC. Proof: In △BDC, By Basic Proportionality Theorem, BOOD=BEEC............(1) But, AOCO=BODO (Given) .........(2) ∴ From (1) and (2) AOCO=BEEC Hence, By Converse of Basic Proportionality Theorem, OE∥AB Now Since, AB∥OE∥DC ∴AB∥DC Hence, ABCD is a trapezium.