QQuestion
ABCD is a circumscribed quadrilateral of a circle with centre O. Show that AB + CD = AD + BC
✓Answer
Given: A quadrilateral ABCD is circumscribed about a circle with centre O.
Let the sides of quadrilateral AB, BC, CD, and DA touch the circle at the points P, Q, R, and S respectively.
To prove: AB + CD = BC + DA
Proof: AS and AP are two tangents to a circle with centre O, drawn from the exterior point A.
∴ AS = AP
Similarly, BP = BQ, CQ = CR, and DR = DS
Now, AB + CD = AP + BP + CR + DR
= AS + BQ + CQ + DS
= (AS + DS) + (BQ + CQ)
= AD + BC
∴ AB + CD = BC + DA (Hence proved)
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