Question

Prove that the lengths of tangents drawn from an external point to a circle are equal. Using the above, prove the following:

A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC

Answer 1

$\angle$OQP and $\angle$ORP are right angles, because these are angles between the radii and tangents,

Now in right triangles $△$ OQP and $△$ ORP,

OQ = OR (Radii of the same circle)

OP = OP (Common)

Therefore, $△$ OQP $\cong$ $△$ ORP (RHS)

This gives PQ = PR

From the above figure we know that

AS = AP

BQ = BP

CQ = CR

DS = DR

Adding the above equations we get

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

AD + BC = AB + CD