# Malfatti Problems

## Document Type

Presentation Abstract

11-13-2001

## Abstract

In 1803 the Italian mathematician Giovanni Malfatti posed the following problem: Given a triangle, find three nonintersecting circles inside of it such that the sum of their areas is maximal. Malfatti and many other mathematicians have thought that the solution of this problem is given by the three circles each of which is tangent to the other two and also to two sides of the triangle. Malfatti has computed the radii of these circles and they are now known as Malfatti's circles. Later on it became clear that the conjecture of Malfatti is not true. Moreover Goldberg proved in 1969 that the Malfatti circles never give a solution of the Malfatti problem, i.e. for any triangle there are three nonintersecting circles inside of it whose area is bigger than the area of the Malfatti circles. As far as the author knows, the Malfatti problem has not been solved yet in the general case although it seems reasonable to conjecture that the solution is given by the greedy algorithm: We first inscribe a circle in the given triangle; then we inscribe a circle in the smallest angle of the triangle which is tangent to the first circle. The third circle is inscribed either in the same angle or in the middle angle of the triangle, depending on which of them has bigger area. In the first part of this lecture we shall discuss the Malfatti problem for two circles in a triangle or in a square. Then we shall consider some problems which, in some sense, are dual to the problems above. Our main purpose is to show how one can solve the Malfatti problem for an equilateral triangle.