The Midpoint Chord

$Based\ on\ the\ properties\ of\ equilateral\ triangle.$ $The\ altitude\ of\ the\ triangle =2Cos30=\sqrt3,\ \therefore\ circumradius,\ R =\frac 2 3 *\sqrt3 \\ and \ chord\ also \ pass\ through\ midpoint\ of\ the\ altitude.\\ \implies\ distance\ of\ the\ chord\ from\ center,\ d =\frac 2 3 *\sqrt3 - \dfrac{\sqrt3} 2=\dfrac 1 {2*\sqrt3}\\ Chord\ length =2*\sqrt{R^2- d^2}=\sqrt 5=2.236.$

Let $L,M$ be the midpoints of the sides of the triangle. Let $X,Y$ be the intersection of the chord with the circle so the order of points on the chord is $XLMY$ .

Clearly $LM=1$ because the smaller triangle is similar to equilateral triangle by $SAS$ and $\frac{2}{2}=1$

Let $XL=x$ . By symettry, $MY=x$ . By intersecting chords at $M$ , $XM \times MY=1 \times 1$ . As $M$ is a midpoint of the triangle.

$XM \times MY=1 \Rightarrow (x+1)(x)=1 \Rightarrow x^2+x-1=0 \Rightarrow x=\frac{-1+\sqrt{5}}{2}$ as $x>0$ .

So $XY=2 \times \frac{-1+\sqrt{5}}{2}+1=-1+\sqrt{5}+1=\sqrt{5}$

So the answer is $\sqrt{5}=2.236$