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| Version | Summary | Created by | Modification | Content Size | Created at | Operation |
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| 1 | Dean Liu | -- | 1732 | 2022-10-09 01:32:20 |
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In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within the triangle. Malfatti's problem has been used to refer both to the problem of constructing the Malfatti circles and to the problem of finding three area-maximizing circles within a triangle. A simple construction of the Malfatti circles was given by (Steiner 1826), and many mathematicians have since studied the problem. Malfatti himself supplied a formula for the radii of the three circles, and they may also be used to define two triangle centers, the Ajima–Malfatti points of a triangle. The problem of maximizing the total area of three circles in a triangle is never solved by the Malfatti circles. Instead, the optimal solution can always be found by a greedy algorithm that finds the largest circle within the given triangle, the largest circle within the three connected subsets of the triangle outside of the first circle, and the largest circle within the five connected subsets of the triangle outside of the first two circles. Although this procedure was first formulated in 1930, its correctness was not proven until 1994.
References
- Ogilvy (1990).
- Wells (1991).
- See also (Ogilvy 1990).
- Andreatta, Bezdek & Boroński (2010).
- Fukagawa & Rothman (2008).
- Simi & Toti Rigatelli (1993).
- (Guy 2007).
- (Paucker 1831); (Zornow 1833); Plücker (1834a, 1834b); (Terquem 1847); (Quidde 1850); (Sylvester 1850); (Scheffler 1851); (Schellbach 1853); Cayley (Cayley|1849}}|1849, Cayley|1854}}|1854, Cayley|1857}}|1857, Cayley|1875–1876}}|1875–1876); (Clebsch 1857); (Talbot 1867); (Wittstein 1871); (Affolter 1873); (Mertens 1873); (Baker 1874); (Schröter 1874); (Simons 1874); (Miller 1875); (Seitz 1875); (Godt 1877); (Lebon 1889); (Bellacchi 1895); (Wedell 1897).
- (Hagge 1908); (Loeber 1914); (Danielsson 1926); (Rogers 1928); (Scardapane 1931); (Procissi 1932); (Eves 1946); (Naitō 1975); (Fiocca 1980); (Hitotumatu 1995); (Takeshima Anai); (Gatto 2000); (Bottema 2001); (Andreatta Bezdek); (Horváth 2014).
- (Casey 1882); (Rouché de Comberousse); (Coolidge 1916); (Baker 1925); (Dörrie 1965); (Ogilvy 1990); (Wells 1991); (Martin 1998); (Andreescu Mushkarov).
- (Hitotumatu 1995); (Takeshima Anai).
- (Martin 1998), exercise 5.20, p. 96.
- According to (Stevanović 2003), these formulae were discovered by Malfatti and published posthumously by him in 1811. However, the 1811 publication, "Résolues", Annales de Mathématiques Pures et Appliquées 1: 347–348, 1811, https://archive.org/stream/annalesdemathma05unkngoog#page/n365/mode/2up , is an unsigned letter (likely from journal editor Joseph Diez Gergonne) giving this formula as equivalent to the results in (Malfatti 1803).
- Miller (1875).
- Weisstein, Eric W.. "Ajima-Malfatti Points". http://mathworld.wolfram.com/Ajima-MalfattiPoints.html. .
- C. Kimberling, Encyclopedia of Triangle Centers , X(179) and X(180). http://faculty.evansville.edu/ck6/encyclopedia/ETC.html
- Encyclopedia of Triangle Centers, X(400).
- (Stevanović 2003).
