#### Question

A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are equal ?

#### Solution

Let ABCD be the quadrilateral circumscribing the circle.

Let E, F, G and H be the points of contact of the quadrilateral to the circle.

**To Prove**: AB + DC = AD + BC**Proof:**

AB = AE + EB

AD = AH + HD

DC = DG + GC

BC = BF + FC

We have:

AE = AH (Tangents drawn from an external point to the circle are equal.)

Similarly, we have:

BE = BF

DH = DG

CG = CF

Now, we have:

AB + DC = AE + EB + DG + GC

= AH + BF + DH + CF

= (AH + DH) + (BF + CF)

= AD + BC

⇒ AB + DC = AD + BC

Thus, if a quadrilateral is drawn to circumscribe a circle, the sums of opposite sides are equal.

Hence, proved.

Is there an error in this question or solution?

Solution A Quadrilateral is Drawn to Circumscribe a Circle. Prove that the Sums of Opposite Sides Are Equal ? Concept: Circles Examples and Solutions.