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© Copyright 2003, Jim Loy
In 1803,
Malfatti showed how to construct, given any right triangle, three circles
inside the triangle and tangent to it and tangent to each other. See the
diagram on the left. This is a fairly difficult construction; I haven't seen
this construction. It can also be done for other triangles. These are called
Malfatti circles.
Furthermore, Malfatti conjectured that
these circles had the greatest possible total area for any non-overlapping
circles inscribed within a triangle. In 1929, it was shown that his conjecture
was not always true. And in 1967, M. Goldberg showed that the conjecture is
never true. Apparently, the circles shown on the right are the real solution to
what is known as the Malfatti problem.
The above diagrams were drawn with the program Cinderella.