Hexagon Triangle Picking
The mean triangle area of a triangle picked inside a regular hexagon with unit area is 
(Woolhouse 1867, Pfiefer 1989). This is a special case of a general polygon triangle picking result due to Alikoski (1939).
The distribution of areas, illustrated above, is apparently not known exactly.
REFERENCES:
Alikoski, H. A. "Über das Sylvestersche Vierpunktproblem." Ann. Acad. Sci. Fenn. 51, No. 7, 1-10, 1939.
Pfiefer, R. E. "The Historical Development of J. J. Sylvester's Four Point Problem." Math. Mag. 62, 309-317, 1989.
Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, p. 114, 1978.
Woolhouse, W. S. B. "Question 2471" Mathematical Questions, with Their Solutions, from the Educational Times, Vol. 8. London: F. Hodgson and Son, pp. 100-105, 1867.
