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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 Steiner (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 this procedure was first formulated in 1930, its correctness was not proven until 1994.

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