Malfatti circles

The three malfatti circuits are circuits of a triangle, of which contacts each of two sides of the triangle and the other two circuits.

Malfatti circles are named after Gianfrancesco Malfatti, who in 1803 stated their construction and expected her to solve the Malfatti problem to pack three circles in a triangle, which do not overlap and have maximum surface area. This was however refuted by 1994 Zalgaller and lot, which showed that the solution is achieved instead by delegating a circle with the largest surface area in each successive steps. That the construction of Malfatti did not solve the Malfatti problem in all cases, already showed praise and Richmond in 1930, and later was even shown that it does so only in the rarest cases. The original Malfatti problem is now known as Malfatti 's marble problem. The entered Malfatti solution of the problem to construct three circles that are touching and two sides of the triangle as a design problem of Malfatti.

For the radii of the Malfatti circles of a triangle ABC the following applies:

Here is the Inkreisradius and for half the triangle perimeter. I is the incenter.

The Malfatti design problem was already solved in a special case of Jakob Bernoulli ( isosceles triangle) and later gave Jakob Steiner (1826, Crelle 's Journal ) using basic geometric and Alfred Clebsch solutions, the latter with elliptic functions (1857, Crelle 's Journal ). Even the Japanese Chokuyen Naonobu Ajima (1732-1798) was a solution within the framework of Japanese architecture.