What is the length of the marked segment?



The marked segment is contained in an equilateral triangle whose side length is \(2-\sqrt{3}\), see .
It is the segment from the midpoint of a side to the center of the triangle. So its length is \(\dfrac{1}{3}\) of the length of the height.
So the marked segment has length \[\dfrac{1}{3} \cdot \dfrac{\sqrt{3}}{2} \cdot (2-\sqrt{3})=\dfrac{2\sqrt{3}-3}{6}\,.\]