A circle touches all the sides of â ABCD. If is the largest side then prove that is the smallest side.

Given that a circle touches all the sides of ABCD and AB is the largest side.

We have to prove that CD is the smallest side.

Proof:

The circle touches all sides of ABCD.

∴ AB + CD = BC + DA … (1)

Given AB is the largest side.

⇒ AB > BC

∴ AB = BC + m

From (1),

⇒ BC + m + CD = BC + DA

⇒ CD + m = DA

∴ CD < DA

Hence CD is smaller than DA. … (2)

But AB is the largest side.

⇒ AB > DA

∴ AB = DA + n

From (1),

⇒ DA + n + CD = BC + DA

⇒ CD + n = BC

∴ CD < BC

Hence CD is smaller than BC. … (3)

AB is largest side, so CD is smaller than AB. … (4)

From (2), (3) and (4), CD is the smallest side of ABCD.