Q. 105.0( 3 Votes )

If all sides of a

Answer :

Consider ABCD as a parallelogram touching the circle at points P, Q, R and S as shown

As ABCD is a parallelogram opposites sides are equal

AB = CD …(a)

AD = BC …(b)

AP and AS are tangents from point A
AP = AS …tangents from point to a circle are equal…(i)

BP and BQ are tangents from point B
BP = BQ …tangents from point to a circle are equal…(ii)

CQ and CR are tangents from point C
CR = CQ …tangents from point to a circle are equal…(iii)

DR and DS are tangents from point D
DR = DS …tangents from point to a circle are equal…(iv)

Add equation (i) + (ii) + (iii) + (iv)

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

From figure AP + BP = AB, CR + DR = CD, AS + DS = AD and BQ + CQ = BC

AB + CD = AD + BC

Using (a) and (b)

AB + AB = AD + AD

2AB = 2AD


AB and AD are adjacent sides of parallelogram which are equal hence parallelogram ABCD is a rhombus

Hence if all sides of a parallelogram touch a circle then that parallelogram is a rhombus

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