सिद्ध कीजिए कि किसी वृत्त के परिगत समांतर चतुर्भुज समचतुर्भुज होता है ।

Explanation

GIVEN A parallelogram `ABCD` circumscribes a circle with centre `O`.
TO PROVE `AB=BC=CD=AD`.
PROOF We know that the lengths of tangents drawn from an exterior point to a circle are equal.
`:. AP=AS`, .......`(i)` [tangents from `A`]
`BP=BQ`, ........`(ii)` [tangents from `B`]
`CR=CQ`, .........`(iii)` [tangents from `C`]
`DR=DS`..........`(iv)` [tangents from `D`]
`:.AB+CD=AP+BP+CR+DR`
`=AS+BQ+CQ+DS` [from `(i)`, `(ii)` `(iii)` and `(iv)`]
`=(AS+DS)+(BQ+CQ)`
`=AD+BC`.
Thus, `AB+CD=AD+BC`
`implies2AB=2AD` [`:'` opposite sides of a `||gm` are equal]
`impliesAB=AD`.
`:.CD=AB=AD=BC`
Hence, `ABCD` is a rhombus