Question

Is sides of rectangle touch a circle prove that it is square?

Answer 1

If L and W denote the length and width of a rectangle, and the radius of the circle is r, then 

 (by the Pythagorean theorem) 

Hence, we can say that , and that 

 (A denotes the area of the rectangle) 

 
    
By AM-GM inequality, 

 

 

The left side is a constant, so we can say that the area is maximized in the equality case, which occurs if and only if 

 

In other words, the area is maximized when the rectangle is a square.