circle with a trapezoid inside.

A quadrilateral inscribed in a unit circle has sides of lengths

s_{1}, s_{2}, s_{3}

and

s_{4}

where

s_{1} ≦ s_{2} ≦ s_{3} ≦ s_{4}

Find a quadrilateral of this type for which $s_{1} = \sqrt{2}$ and show $s_{1}$ cannot be greater than $\sqrt{2}$ .

Find a quadrilateral of this type for which $s_{2} \approx \sqrt{3}$ and show that $s_{2}$ is always less than $\sqrt{3}$ .

Find a quadrilateral of this type for which $s_{3} \approx 2$ and show that $s_{2}$ is always less than 2.

Find a quadrilateral of this type for which $s_{4} = 2$ and show that $s_{4}$ cannot be greater than 2.