The problem is from Kiselev's Geometry exercise 317.
Through two given points on a circle, construct two parallel chords with a given sum.
Here is what I have tried so far:
Mark the two points by $A$ and $C$ respectively. If we have constructed such two chords and marked the two other points by $B$ and $D$, the quadrilateral $ABCD$ is an isosceles trapezoid where $AC$ is a diagonal and (without loss of generality) $AB$ and $CD$ are parallel. The midline of the bases measures half of the given sum, and it passes through the midpoint of the diagonal $AC$.
Unfortunately, I could not progress any further from here; I think I should utilize the fact that the 4 points are concyclic and $ABCD$ is an isosceles trapezoid, but I could not find usage of the fact.
Any help would be much appreciated.