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). Ogilvy (1990,
pp. 146-147) and Wells (1991) illustrate specific cases where alternative solutions
are clearly optimal.
The general Malfatti problem on an arbitrary triangle was actually formulated and solved earlier by the Japanese geometer Chokuen Ajima (1732-1798) (Fukagawa and Pedoe
1989, p. 28; Kimberling). It asks to draw within a given triangle
three circles, each of which is tangent
to the other two and to two sides of the triangle. The
resulting circles so constructed (tangent to and ), ( and ), and (tangent to and ) are known as the Malfatti
circles. The problem was solved using an algebraic-geometric solution by Malfatti
(1803; Ostwald; Dörrie 1965, p. 147), and a purely geometric solution was
given without proof by Steiner (1826; Ostwald; Dörrie 1965, p. 147).
The Malfatti configuration appears on the cover of Martin (1998).
(tangent to 
and 
), 
(
and 
), and 
(tangent to 
and 
) are known as the Malfatti
circles. The problem was solved using an algebraic-geometric solution by Malfatti
(1803; Ostwald; Dörrie 1965, p. 147), and a purely geometric solution was
given without proof by Steiner (1826; Ostwald; Dörrie 1965, p. 147).
