A quadrilateral ABCD is drawn to circumscribe a circle, as shown in the figure. Prove that AB + CD = AD + BC
We known 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) [using (i), (ii), (iii), (iv)]
= (AS + DS) + (BQ + CQ)
= (AD + BC).
Hence, (AB + CD) = (AD + BC)