if abcd is a quadrilateral and a circle is inscribed in it prove ab +cd = ad + bc

Quadrilateral ABCD circumscribes the circle. 

∴ AB, BC, CD and DA are tangents to the circle at the point of contact P, Q, R and S respectively.

As, we know that, the lengths of tangents drawn from an external point to a circle are equal.

⇒AP = AS ... (1)

 BP = BQ ... (2)

 CR = CQ ... (3)

 DR = DS ... (4)

Now, on adding (1), (2), (3) and (4), we get,

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

⇒(AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)

⇒AB + CD = AD + BC