What is the probability that a randomly selected chord of the unit circle is longer than one side of an equilateral triangle inscribed in this circle?

version 1

Each chord can be clearly identified by its center point (the chord is then formed normal to the line connecting the circle center point). The center point P of the chord is given by the Cartesian coordinates (x, y). The event space Ω1 is given in this way by. Event A1: The selected chord is larger than the side of the triangle; The probability for A1 amounts .task Move the Point p and the upper point of the triangleto familiarize yourself with the problem. Display the simulation.

Variant 2

The center of the chord can also be given by its polar coordinates (r; φ). The event space Ω2 is given in this way by. Event A2: The selected chord is larger than the side of the triangle; The probability for A2 amounts .task Move the Point p and the top tip of the triangle. Display the simulation.

Variation 3

The chord can also be given by a point on the edge of the circle (i.e. by the arc length b) and an angle α to the tangent. The event space Ω3 is given in this way by. Event A3: The selected chord is larger than the side of the triangle; The probability for A3 amounts . task Move the Point p and the upper point of the triangle as well as the slider for the angle α. Display the simulation.
The Paradox is that there are different probabilities depending on the choice of tendon. The reason for this lies in the fact that the random selection of a tendon is not precisely specified.