A circle touches all the four sides of a quadrilateral ABCD. Prove tha

1.	A circle touches all the four sides of a quadrilateral ABCD. Prove thatAB +CD=BC+DA
Answer» From the figure we observe that, DR = DS (Tangents on the circle from point D) … (i) AP = AS (Tangents on the circle from point A) … (ii) BP = BQ (Tangents on the circle from point B) … (iii) CR = CQ (Tangents on the circle from point C) … (iv) Adding all these equations, DR + AP + BP + CR = DS + AS + BQ + CQ ⇒ (BP + AP) + (DR + CR) = (DS + AS) + (CQ + BQ) ⇒ CD + AB = AD + BC

A circle touches all the four sides of a quadrilateral ABCD. Prove thatAB +CD=BC+DA

Answer»

From the figure we observe that,

DR = DS (Tangents on the circle from point D) … (i)

AP = AS (Tangents on the circle from point A) … (ii)

BP = BQ (Tangents on the circle from point B) … (iii)

CR = CQ (Tangents on the circle from point C) … (iv)

Adding all these equations,

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

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

⇒ CD + AB = AD + BC