There is no paradox, any methods conceived to generate chords must generate all chords possible in the circle. Otherwise a subset of the infinite set of chords possible will give a wrong answer.
Methods that are off the circle center cannot use symmetry to generate all chords with the same required condition, \(\bar{c}\gt\bar{a}\). Focusing on the sides and vertices of the triangle does not provide the symmetry that generate all possible chords as the condition 'any random chords, c' requires. The triangle has to be rotated also but it does not have circular symmetry. Choosing any sides or vertices introduces conditional probabilities that must be resolved.
Given any point in the circle, infinite number of chords through it is possible. Methods that draw only one chord through each point have ignored many others through the same point that meet the required condition, \(\bar{c}\gt\bar{a}\). Setting the chosen point as midpoint of one chord added extra restrain/condition to the problem and reduced the set of chords considered.
In particular, the center of the circle can have more than one chord through it that is longer than the side of the triangle. Missing out on these chords, undercounts the long chords and so gives a lower probability for long chords.
OK?
Note: The rest of the triangle does not matter, only the length \(a\) and the parameter that defines \(a\); \(120^o\) at the center of the circle.
