Question

In figure a quadilateral ABCDis drawn to circumscribe a circle with centre O,in such a way that the sides AB,BC,CD and DA touch the circle at the points P,Q,R and S respectively.Prove that AB+CD=BC+DA

Answer 1

AB+CD= BC+DA
AB= AP+PB
CD=DR+CR
( tangent property) (AP=AS, PB=BQ,DR=DS, CR=CQ)
therefore, AB+CD= AP+PB+DR+CR
=AS+BQ+DS+CQ
=(AS+DS)+(BQ+CQ)
=DA+BC
hence proved, AB+CD=BC+DA