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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).

See also

Ajima-Malfatti Points, Circle Packing, Malfatti Circles, Marble Problem, Tangent Circles

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References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin, Ireland: Hodges, Figgis, & Co., pp. 154-155, 1888.Dörrie, H. "Malfatti's Problem." §30 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 147-151, 1965.Eves, H. A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, p. 245, 1965.F. Gabriel-Marie. Exercices de géométrie. Tours, France: Maison Mame, pp. 710-712, 1912.Forder, H. G. Higher Course Geometry. Cambridge, England: Cambridge University Press, pp. 244-245, 1931.Fukagawa, H. and Pedoe, D. "The Malfatti Problem." Japanese Temple Geometry Problems (San Gaku). Winnipeg, Manitoba, Canada: The Charles Babbage Research Centre, pp. 28 and 103-106, 1989.Gardner, M. Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 163-165, 1992.Goldberg, M. "On the Original Malfatti Problem." Math. Mag. 40, 241-247, 1967.Hart. Quart. J. 1, p. 219.Kimberling, C. "1st and 2nd Ajima-Malfatti Points." https://faculty.evansville.edu/ck6/tcenters/recent/ajmalf.html.Lob, H. and Richmond, H. W. "On the Solution of Malfatti's Problem for a Triangle." Proc. London Math. Soc. 2, 287-304, 1930.Malfatti, G. "Memoria sopra un problema stereotomico." Memorie di matematica e fisica della Societé Italiana delle Scienze 10-1, 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.Oswald. Klassiker de exakten Wissenschaften, Vol. 23. Suppl.Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.Rouché, E. and de Comberousse, C. Traité de géométrie plane. Paris, France: Gauthier-Villars, pp. 311-314, 1900.Schellbach. J. reine angew. Math. 45.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, 1991.Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, pp. 206-209, 1961.

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Malfatti's Problem

Cite this as:

Weisstein, Eric W. "Malfatti's Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MalfattisProblem.html

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