Median Area for Broken Sticks

Breaking a line segment L in two places at random, the three pieces can be configured as a triangle T with probability 1/4. We determine both the PDF and CDF for area(T) in terms of elliptic integrals. In particular, if L has length 1, then the median area 0.031458… can be calculated to arbitrary precision. We also mention the analog involving cyclic quadrilaterals – with corresponding probability 1/2 – and ask some unanswered questions.