A circle is inscribed in a square with sides = 40.
A smaller (of course!) circle tangent to the above
circle and 2 sides of the square is inscribed in
one of the corners of the square.
What is the diameter of this circle?
The radius of the large circle plus the diameter of the smaller corner circle is equal to the half the length of the diagonal of the square.
$R + d = 20\sqrt{2}$
$R$ of course is just half the length of the side of the square.
$R=\dfrac{40}{2}=20$
$20+d = 20\sqrt{2}$
$d = 20(\sqrt{2}-1)\approx 8.3$