Malfatti circles

In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within the triangle.

Malfatti's problem has been used to refer both to the problem of constructing the Malfatti circles and to the problem of finding three area-maximizing circles within a triangle. A simple construction of the Malfatti circles was given by Jakob Steiner in 1826, and many mathematicians have since studied the problem. Malfatti himself supplied a formula for the radii of the three circles, and they may also be used to define two triangle centers, the Ajima–Malfatti points of a triangle.

The problem of maximizing the total area of three circles in a triangle is never solved by the Malfatti circles. Instead, the optimal solution can always be found by a greedy algorithm that finds the largest circle within the given triangle, the largest circle within the three connected subsets of the triangle outside of the first circle, and the largest circle within the five connected subsets of the triangle outside of the first two circles. Although it was first formulated in 1930, the correctness of this procedure was not proven until 1994.

In 1803 Gian Francesco Malfatti posed the problem of cutting three cylindrical columns out of a triangular prism of marble, maximizing the total volume of the columns. He assumed, as did many others after him, that the solution to this problem was given by three tangent circles within the triangular cross-section of the wedge. That is, more abstractly, he conjectured that the three Malfatti circles have the maximum total area of any three disjoint circles within a given triangle.

Malfatti published in Italian and his work may not have been read by many in the original. It was popularised for a wider readership in French by Joseph Diaz Gergonne in the first volume of his Annales (1810/11), with further discussion in the second and tenth. However, this advertisement most likely acted as a filter, as Gergonne only stated the circle-tangency problem, not the area-maximizing one.

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