Given:
- ABCD is a quadrilateral.
- A circle touches all the sides of quadrilateral ABCD.
- AB is the largest side.

To prove:
- CD is the smallest side.

Proof:
Let us assume that AB is the largest side and CD is not the smallest side.
Since the circle touches all the sides of the quadrilateral, the circle must be tangent to each side of the quadrilateral at a point.

Case 1: CD is the largest side
If CD is the largest side, then the circle would be tangent to CD at a point, but it would not be tangent to any other side of the quadrilateral. This is not possible as the circle must touch all the sides of the quadrilateral. Therefore, CD cannot be the largest side.

Case 2: BC is the largest side
If BC is the largest side, then the circle would be tangent to BC at a point. Since AB is the largest side, the circle must be tangent to AB at a point as well. But this creates a contradiction because the circle cannot be tangent to two non-parallel sides of the quadrilateral. Therefore, BC cannot be the largest side.

Case 3: AD is the largest side
If AD is the largest side, then the circle would be tangent to AD at a point. Since AB is the largest side, the circle must be tangent to AB at a point as well. But this creates a contradiction because the circle cannot be tangent to two non-parallel sides of the quadrilateral. Therefore, AD cannot be the largest side.

Case 4: AB is the largest side
If AB is the largest side, then the circle would be tangent to AB at a point. Since the circle touches all the sides of the quadrilateral, it must also be tangent to BC, CD, and DA. In this case, CD must be the smallest side because it is tangent to the circle at a point and the other sides are not tangent to the circle.

Therefore, it is proved that if a circle touches all the sides of a quadrilateral and AB is the largest side, then CD is the smallest side.