ABSTRACT
What is the chance that a random chord of a circle is longer than the side of an inscribed equilateral triangle?
The chords from a vertex of the triangle to the circumference are longer if they lie within the angle at the vertex. Since that is true of one-third of the chords, the probability is one-third.
The chords parallel to one side of such a triangle are longer if they intersect the inner half of the radius perpendicular to them, so that their midpoint falls within the triangle. So the probability is one-half.
A chord is also longer if its midpoint falls within a circle inscribed within the triangle. The inner circle will have a radius one-half and therefore an area one-quarter that of the outer one. So the probability is one-quarter.
