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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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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Distributing production has been his profession for some time. Ohio is where his home is and his family loves it. To play lacross is the thing I adore most of all.
Review my page: email psychic readings (breenq.com)