# Malfatti circles

In 1803 the Italian mathematician Giovanni Malfatti (1731–1807) posed
the following problem: Given a triangle,
find three non-overlapping circles inside
it such that the sum of their areas is maximal. Malfatti and many other
mathematicians thought that the solution is given by the three circles each
of which is tangent to the other two and
also to two sides of the triangle. Malfatti computed the radii of these
circles, and they are now known as **Malfatti's circles**.

Later it became clear that the conjecture of Malfatti is not true. In particular,
Goldberg proved in 1969 that the Malfatti circles *never* give a
solution of the Malfatti problem! In other words, for any triangle there
are three non-intersecting circles inside it, whose areas are bigger than
the area of the Malfatti circles. So far as is known, the Malfatti problem
hasn't been solved yet in the general case although it seems reasonable
to conjecture that the solution is given by what is called 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 the
bigger area.