Cutting the sides

Let the circle be centred at $O=(0,0)$ . We also know that the circle has radius $r=\frac{2}{\sqrt{3}}$ . Thus, the equation of the circle is $x^2+y^2=\frac{4}{3}$ Now, we can construct points $A(0,\frac{2}{\sqrt{3}}),B(-1,-\frac{1}{\sqrt{3}}),C(1,-\frac{1}{\sqrt{3}})$ Let $E,F$ be the midpoint of $AB,AC$ respectively. Then, using the midpoint theorem, we find that $E=(-\frac{1}{2},\frac{1}{2\sqrt{3}}),F=(\frac{1}{2},\frac{1}{2\sqrt{3}})$ The equation of $EF$ is thus $y=\frac{1}{2\sqrt{3}}$ . Putting it back into the equation of the circle, we have $x^2+\frac{1}{12}=\frac{4}{2}\\x^2=\frac{5}{4}\\x=\frac{\sqrt{5}}{2}$ The length of the chord is $2x=\sqrt{5}=2.236$