Malfatti Circles

Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles. The Malfatti configuration appears on the cover of Martin (1998).

The positions and radii of the Malfatti circles can be found by labeling sides and distances as illustrated above. The length of the projection of the line connecting circles and onto side can be found from the diagram at right to be

			(1)
			(2)

Therefore, three equations follow from the condition that the labeled lengths must sum to the side lengths,

			(3)
			(4)
			(5)

Three additional equations follow from the fact that the circle centers lie on the corresponding angle bisectors of the triangle vertices, so

			(6)
			(7)
			(8)

Re-expressing these equations in terms of side lengths and rearranging and squaring to eliminate square roots then gives the system of six polynomial equations

	(9)
	(10)
	(11)
	(12)
	(13)
	(14)

This system can be solved simultaneously for the radii and distances. The radius and position of the -circle is given by appropriate roots of the complicated resulting polynomial

(15)

In particular,

			(16)
			(17)
			(18)

where is a polynomial root, and the centers are given by

			(19)
			(20)
			(21)

Let the circles have radii , , and . Then the inradius of the triangle in which these circles are inscribed is given by

			(22)
			(23)

(Fukagawa and Pedoe 1989, p. 106).

Although these circles were for many years thought to provide the solutions to Malfatti's problem, they were subsequently shown never to provide the solution.