The Bertrand paradox (probability) explains Bertrand paradox in probability.
In "Random Midpoint method" Bertrand uses a concept that all chords whose midpoints lie in the inscribed radius of equilateral triangle have length greater than side of triangle.
How to prove this?
Rotate the triangle so that one vertex coincides with one endpoint of the chord. Then you see (sketch!) that the central angle over the chord is larger than that over the side.