Malfatti Circles - History

History

The problem of constructing three circles tangent to each other within a triangle was posed by the 18th-century Japanese mathematician Ajima Naonobu prior to the work of Malfatti, and included in an unpublished collection of Ajima's works made a year after Ajima's death by his student Kusaka Makoto. Even earlier, the same problem was considered in a 1384 manuscript by Gilio di Cecco da Montepulciano, now in the Municipal Library of Sienna, Italy.

Since the work of Malfatti, there has been a significant amount of work on methods for constructing Malfatti's three tangent circles; Richard K. Guy writes that the literature on the problem is "extensive, widely scattered, and not always aware of itself". Notably, in 1826 Jakob Steiner presented a simple geometric construction based on bitangents; other authors have since claimed that Steiner's presentation lacked a proof, which was later supplied by Andrew Hart (1856), but Guy points to the proof scattered within two of Steiner's own papers from that time. Lob and Richmond cite solutions by C. L. Lehmus (1819), Eugène Charles Catalan (1845), J. Derousseau (1895), A. Pampuch (1904), and J. L. Coolidge (1916), all based on algebraic formulations of the problem. The algebraic solutions do not distinguish between internal and external tangencies among the circles and the given triangle; if the problem is generalized to allow tangencies of either kind, then a given triangle will have 32 different solutions and conversely a triple of mutually tangent circles will be a solution for eight different triangles. Bottema (2001) and Guy (2007) cite additional work on the problem and its generalizations by C. Adams (1846), Adolphe Quidde (1850), K. H. Schellbach (1853), Arthur Cayley (1854, 1857, 1875), Alfred Clebsch (1857), P. Simons (1874), J. Casey (1888), Rouché and Comberousse (1900), H. F. Baker (1925), L. J. Rogers (1928), Angelo Procissi (1932), Jun Naito (1975), and D. G. Rogers (2005).

Gatto (2000) and Mazzotti (1998) recount an episode in 19th-century Neapolitan mathematics related to the Malfatti circles. In 1839, Vincenzo Flauti, a synthetic geometer, posed a challenge involving the solution of three geometry problems, one of which was the construction of Malfatti's circles; his intention in doing so was to show the superiority of synthetic to analytic techniques. Despite a solution being given by Fortunato Padula, a student in a rival school of analytic geometry, Flauti awarded the prize to his own student, Nicola Trudi, whose solutions Flauti had known of when he posed his challenge. More recently, the problem of constructing the Malfatti circles has been used as a test problem for computer algebra systems.