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Find the maximum total Area of three Circles (of possibly different sizes) which can be packed inside a Right Triangle of any shape without overlapping. In 1803, Malfatti gave the solution as three Circles (the Malfatti Circles) tangent to each other and to two sides of the Triangle. In 1929, it was shown that the Malfatti Circles were not always the best solution. Then Goldberg (1967) showed that, even worse, they are never the best solution.
See also Malfatti's Tangent Triangle Problem
References
Eves, H. A Survey of Geometry, Vol. 2. Boston: Allyn & Bacon, p. 245, 1965.
Goldberg, M. ``On the Original Malfatti Problem.'' Math. Mag. 40, 241-247, 1967.
Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 145-147, 1990.