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.
*) For five years the speaker has been the coach
of the Bulgarian Mathematics Olympic Team, which constantly has
been successful at the International Mathematical Olympiads for
high school students. He will share his experience on working and
interaction with gifted students.
