What is the length of the marked segment?



The marked segment is the side of a triangle.
This triangle is equilateral.
Indeed, one angle is in common with the big equilateral triangle while the other two are exterior angles of a regular hexagon.
As this triangle and the hexagon have one side in common, they have the same side length.
So three segments of equal length (overlapping only in two points) cover the side of the big triangle.
The side of the big triangle has length \(1\) hence the length of the marked segment is \(\dfrac{1}{3}\).