Seems unlikely, because the density of chords is much higher near the center of the circle than the rim. (The density of chords is infinite at exactly the cente

Seems unlikely, because the density of chords is much higher near the center of the circle than the rim. (The density of chords is infinite at exactly the center). This combines with the fact the points are farther apart when their bisector is closer to the center, making it harder for all 4 of them to be on a half-circumference.