Question

A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC

Fig. 10.12

Answer 1

It can be observed that
DR = DS (Tangents on the circle from point D)        .....… (1)
CR = CQ (Tangents on the circle from point C)        ......… (2)
BP = BQ (Tangents on the circle from point B)         …..… (3)
AP = AS (Tangents on the circle from point A)         …..… (4) 
Adding all these equations, we obtain
DR + CR + BP + AP = DS + CQ + BQ + AS
(DR + CR) + (BP + AP) = (DS + AS) + (CQ + BQ)
CD + AB = AD + BC