Circular uncertainties

A chord is drawn in a circle in a random way. What is the probability that its length is greater than the side of an inscribed equilateral triangle ?

One way to calculate that would be to choose, for reasons of symmetry, the direction of the chord and to consider its intersection with the perpendicular diameter. The chord is of maximum length when it intersects the diameter in the middle of the radius. The probability in question is therefore 1/2.

Another method would be to consider the midpoint of the chord. This midpoint must lie within aconcentric circle of half the radius. Since the area of the new circle is one quarter as large, the probability in question is V4. Joseph Bertrand proposed this problem half a century ago as a means of criticizing continuous probabilities and added a third method of calculation that leads to a result contradicting the two preceding results. What is this method ?

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