Question

Prove that a rectangle circumscribing a circle is a square

Answer 1

In the provided figure the marking are done by the rule that 'the two tangents drawn to a circle from a fixed point has same length'

As the quadrilateral is a rectangle
       a+d=b+c
   ∴ a=b+c-d-----------------------------------(1)

From the above reason
      a+b=c+d
   from (1)
       b+c-d+b=c+d
   ⇒ 2b=2d
   ⇒ b=d-----------------------------------------(2)

Substituting from (2) in (1)
       a+d=c+d
   ⇒ a=c------------------------------------------(3)
  
   From (2) and (3)
      a+d = a+b-----------------------------------(4)
  Similarly we can prove that
      a+b=b+c-----------------------------------(5)
and      b+c=c+d------------------------------------(6)
and      c+d=a+d------------------------------------(7)
From (4) , (5) , (6) and (7) we can prove that the rectangle is a square.
    

Answer 2

In the provided figure the marking are done by the rule that 'the two tangents drawn to a circle from a fixed point has same length'

As the quadrilateral is a rectangle

       a+d=b+c

   ∴ a=b+c-d-----------------------------------(1)

From the above reason

      a+b=c+d

   from (1)

       b+c-d+b=c+d

   ⇒ 2b=2d

   ⇒ b=d-----------------------------------------(2)

Substituting from (2) in (1)

       a+d=c+d

   ⇒ a=c------------------------------------------(3)

  

   From (2) and (3)

      a+d = a+b-----------------------------------(4)

  Similarly we can prove that

      a+b=b+c-----------------------------------(5)

and      b+c=c+d------------------------------------(6)

and      c+d=a+d------------------------------------(7)

From (4) , (5) , (6) and (7) we can prove that the rectangle is a square.