Question

If the sides of a rectangle touch a circle, prove that the rectangle is a square.

Answer 1

A rectangle with all sides touching a circle implies the circle is an incircle.

To Prove

A rectangle whose four sides touch a circle is a square.

A rectangle with all sides touching a circle implies that the circle is an incircle of the rectangle.

Let the rectangle be ABCD, with length l and breadth b.

A circle touches all four sides at points P, Q, R, and S.

In any quadrilateral that has an incircle,

the sum of the lengths of opposite sides is equal:

AB + CD = BC + AD.

For rectangle ABCD:

AB = CD = l,

BC = AD = b.

l + l  =b + b

2l = 2b

l = b

Since the length and breadth are equal, the rectangle is a square.

AB = CD = l