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| [[File:Orthic triangle.png|thumb|380px|orthic triangle: <math>\triangle DEF </math> <br/> inscribed triangles: <math>\triangle DEF\,,\triangle GHI </math> <br/> <math>|DE|+|EF|+|FD|\leq |GH|+|HI|+|IG| </math>]]
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| In [[geometry]], '''Fagnano's problem''' is an [[Optimization (mathematics)|optimization]] problem that was first stated by [[Giovanni Fagnano]] in 1775:
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| :''For a given [[acute triangle]] determine the inscribed triangle of minimal [[perimeter]]''.
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| The [[orthic 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]] methods and an intermediate result given by his father [[Giulio Carlo de' Toschi di Fagnano]]. Later however several geometric proofs were discovered as well, amongst others by [[Hermann Schwarz]] and [[Lipót Fejér]]. These proofs use the geometrical properties of reflections to determine some minimal path representing the perimeter.
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| ==References==
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| *Heinrich Dörrie: ''100 Great Problems of Elementary Mathematics: Their History and Solution''. Dover Publications 1965, ISBN 0-486-61348-8, problem 90 ([http://books.google.de/books?id=i4SJwNrYuAUC&pg=PA359&vq=fagnano&source=gbs_search_r&cad=1_1 restricted online version (Google Books)])
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| *Paul J. Nahin: ''When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible''. Princeton University Press 2004, ISBN 0-691-07078-4, p. 67
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| *[[Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]]; Greitzer, S. L.:''Geometry Revisited''. Washington, DC: Math. Assoc. Amer. 1967, pp. 88–89.
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| ==External links==
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| *[http://www.cut-the-knot.org/Curriculum/Geometry/Fagnano.shtml Fagnano's problem at cut-the-knot]
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| *[http://eom.springer.de/f/f038140.htm Fagnano's problem] in the [[Encyclopaedia of Mathematics]]
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| *[http://www.pballew.net/orthocen.html Fagnano's problem at a website for triangle geometry]
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| *{{MathWorld|urlname=FagnanosProblem|title=Fagnano's problem}}
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| {{DEFAULTSORT:Fagnano'S Problem}}
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| [[Category:Triangle geometry]]
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| [[Category:Mathematical problems]]
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| {{elementary-geometry-stub}}
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