Construct three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle.
In 1803 Gian Francesco Malfatti conjectured that solving this problem would provide a solution to Malfatti's problem or the marble problem, of how to cut three marble columns of maximal area out of a triangular wedge of marble. The conjecture is wrong; that some solutions are not optimal was shown in 1930, while Goldberg showed that none of the Malfatti circles are ever optimal in 1967 (Ogilvy 1990). Ogilvy (1990) and Wells (1991) then illustrated specific cases where alternative solutions are clearly optimal. In 1992, V.A. Zalgaller and G.A. Los' gave a complete solution of the marble problem.
* Dörrie, H. "Malfatti's Problem." §30 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 147-151, 1965. ISBN 0-486-61348-8
* Goldberg, M. "On the Original Malfatti Problem." Math. Mag. 40, 241-247, 1967.
* Ogilvy, C. S. Excursions in Geometry. New York: Dover, 1990. ISBN 0-486-26530-7
* Eric W. Weisstein, Malfatti Circles at MathWorld.
* Eric W. Weisstein, Malfatti's Problem at MathWorld.
* Malfatti's Problem at cut-the-knot
* Zalgaller, V.A. and Los', G.A. (1994). "The solution of Malfatti's problem". Journal of Mathematical Sciences 72 (4): 3163–3177. Springer.