geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, Fagnano's problem is an

optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

problem that was first stated by

Giovanni Fagnano Giovanni Francesco Fagnano dei Toschi (born 31 January 1715 in Senigallia, died 14 May 1797 in Senigallia) was an Italian churchman and mathematician, the son of Giulio Carlo de' Toschi di Fagnano, also a mathematician. Religious career Fagnano was ...

in 1775: The solution is the orthic triangle, with vertices at the base points of the

altitudes Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...

of the given triangle.

Solution

The orthic triangle, with vertices at the base points of the

of the given triangle, has the smallest perimeter of all triangles inscribed into an acute triangle, hence it is the solution of Fagnano's problem. Fagnano's original proof used

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

methods and an intermediate result given by his father

Giulio Carlo de' Toschi di Fagnano Giulio Carlo, Count Fagnano, Marquis de Toschi (26 September 1682 — 18 May 1766) was an Italian mathematician. He was probably the first to direct attention to the theory of elliptic integrals. Fagnano was born in Senigallia (at the time spel ...

. Later however several geometric proofs were discovered as well, amongst others by

Hermann Schwarz Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis. Life Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kummer, ...

and

Lipót Fejér Lipót Fejér (or Leopold Fejér, ; 9 February 1880 – 15 October 1959) was a Hungarian mathematician of Jewish heritage. Fejér was born Leopold Weisz, and changed to the Hungarian name Fejér around 1900. Biography Fejér studied mathematic ...

. These proofs use the geometrical properties of reflections to determine some minimal path representing the perimeter.

Physical principles

A solution from physics is found by imagining putting a rubber band that follows

Hooke's Law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...

around the three sides of a triangular frame

ABC

, such that it could slide around smoothly. Then the rubber band would end up in a position that minimizes its elastic energy, and therefore minimize its total length. This position gives the minimal perimeter triangle. The tension inside the rubber band is the same everywhere in the rubber band, so in its resting position, we have, by Lami's theorem,

\angle bcA = \angle acB, \angle caB = \angle baC, \angle abC = \angle cbA

Therefore, this minimal triangle is the orthic triangle.

References

*Heinrich Dörrie: ''100 Great Problems of Elementary Mathematics: Their History and Solution''. Dover Publications 1965, p. 359-360. , problem 90
restricted online version (Google Books)
*

Paul J. Nahin Paul J. Nahin (born November 26, 1940 in Orange County, California) is an American electrical engineer and author who has written 20 books on topics in physics and mathematics, including biographies of Oliver Heaviside, George Boole, and Claude Sh ...

: ''When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible''. Princeton University Press 2004, , p. 67 * Coxeter, H. S. M.; Greitzer, S. L.:''Geometry Revisited''. Washington, DC: Math. Assoc. Amer. 1967, pp. 88–89. * H.A. Schwarz: ''Gesammelte Mathematische Abhandlungen, vol. 2''. Berlin 1890, pp. 344-345.
online
at the

Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music ...

, German)

External links

Fagnano's problem at cut-the-knotFagnano's problem
in the

Encyclopaedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a gradua ...

Fagnano's problem at a website for triangle geometry
* {{DEFAULTSORT:Fagnano'S Problem Triangle problems

Solution

Physical principles

See also

References

External links