The one Triangle in addition to the Equilateral Triangle for which the largest inscribed Square
can be inscribed in three different ways. The ratio of the sides to that of the base is given by
(Sloane's A046095), where
See also Graham's Biggest Little Hexagon
References
Conway, J. H. and Guy, R. K. ``Calabi's Triangle.'' In The Book of Numbers. New York:
Springer-Verlag, p. 206, 1996.
Sloane, N. J. A.
A046095 and
A046096
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html.