In 1803, Malfatti posed the problem of determining the three circular columns of marble of possibly different sizes which, when carved out of a right triangular prism,
would have the largest possible total cross section.
This is equivalent to finding the maximum total area of
three circles which can be packed inside a right
triangle of any shape without overlapping. This problem is now known as the marble
problem (Martin 1998, p. 92).

Malfatti gave the solution as three circles (the Malfatti circles) tangent to each other and to two sides of the triangle.
In 1930, 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 (Ogilvy 1990, pp. 145-147).

Zalgaller and Los' (1994) give analytic conditions for the solution of the marble problem.

Goldberg, M. "On the Original Malfatti Problem." Math. Mag.40, 241-247, 1967.Malfatti, G. "Memoria
sopra un problema stereotomico." Memorie di matematica e fisica della Societé
Italiana delle Scienze10-1, 235-244, 1803.Martin, G. E.
Geometric
Constructions. New York: Springer-Verlag, pp. 92-95, 1998.Ogilvy,
C. S. Excursions
in Geometry. New York: Dover, 1990.Zalgaller, V. A. and Los',
G. A. "Solution of the Malfatti Problem." Ukrain. Geom. Sb.,
No. 35, 14-33 and 161, 1992. English translation in J. Math. Sci.72,
3163-3177, 1994.