Question

If the sides of a quadrilateral ABCD touch a circle, then AB+CD=BC+AD

• A. True
• B. False

Answer 1

The correct option is A

True

Given - A circle touches the sides AB, BC, CD and DA of quad. ABCD at P,Q, R, and S respectively.

To Prove - AB+ CD = BC+AD
Proof - Since AP and AS are the tangents to the circle from external points A.
$\therefore$ AP=AS ....(i)
Similarly, we can prove that,
BP = BQ ....(ii)
CR=CQ ....(iii)
DR = DS ....(iv)
Adding, we get:
AP+BP+CR+DR=AS+BQ+CQ+DS
$\Rightarrow$ AP+BP+CR+DR=AS+DS+BQ+CQ
AB+CD=AD+BC

Hence AB+CD=BC+AD