Question 1
Given the equilateral triangle $ABC$, side $a$, and its circumscribed circumference, is considered the circle segment limited by the string $AB$ and the arc (of $120^{\circ}$) with the same ends. When cutting this circular segment with lines parallel to the side $BC$, a segment of points inside the mentioned circular segment is determined on each of them. Determine the maximum length of those rectilinear segments.