What is the length of the marked segment?



The marked segment is a side of the square and a side of one of the equilateral triangles.
Let \(s\) be its length.
The height of the triangles is then \(\dfrac{\sqrt{3}}{2}s\).
Consider a diameter of the circle connecting two triangle vertices on the circumference.

This diameter can be subdivided into three segments: two triangle heights and one segment whose length is \(s\).
We deduce that \[2=s+2\cdot (\dfrac{\sqrt{3}}{2}s).\] The side length of the square is then \[\sqrt{3}-1.\]