Q. 174.0( 3 Votes )

A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are equal.

Answer :


Let us consider a quadrilateral ABCD, And a circle is circumscribed by ABCD


Also, Sides AB, BC, CD and DA touch circle at P, Q, R and S respectively.


To Proof : Sum of opposite sides are equal, i.e. AB + CD = AD + BC


Proof :


In the Figure,


As tangents drawn from an external point are equal.


We have


AP = AS


[tangents from point A]


BP = BQ


[tangents from point B]


CR = CQ


[tangents from point C]


DR = DS


[tangents from point D]


Add the above equations


AP + BP + CR + DR = AS + BQ + CQ + DS


AB + CD = AS + DS + BQ + CQ


AB + CD = AD + BC


Hence Proved.


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