A quadrilateral ABCD is drawn to circumscribe a circle, as shown in the given figure. Prove that: AB + CD = AD + BC
Given: A quadrilateral ABCD, And a circle is circumscribed by ABCD
Also, Sides AB, BC, CD and DA touch circle at P, Q, R and S respectively.
To Prove: AB + CD = AD + BC
Proof:
In the Figure,
As tangents drawn from an external point are equal.
We have
AP = AS [tangents from point A]
BP = BQ [tangents from point B]
CR = CQ [tangents from point C]
DR = DS [tangents from point D]
Add the above equations
⇒ AP + BP + CR + DR = AS + BQ + CQ + DS
⇒ AB + CD = AS + DS + BQ + CQ
⇒ AB + CD = AD + BC
Hence Proved.