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339:, 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
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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
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When Least is Best: How
Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible
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440:{\displaystyle \angle bcA=\angle acB,\angle caB=\angle baC,\angle abC=\angle cbA}
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A solution from physics is found by imagining putting a rubber band that follows
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100 Great
Problems of Elementary Mathematics: Their History and Solution
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530:. Washington, DC: Math. Assoc. Amer. 1967, pp. 88–89.
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Therefore, this minimal triangle is the orthic triangle.
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methods and an intermediate result given by his father
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Fagnano's problem at a website for triangle geometry
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219:{\displaystyle |DE|+|EF|+|FD|\leq |GH|+|HI|+|IG|}
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538:Gesammelte Mathematische Abhandlungen, vol. 2
313:around the three sides of a triangular frame
91:{\displaystyle \triangle DEF\,,\triangle GHI}
253:determine the inscribed triangle of minimal
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16:Optimisation problem in triangle geometry
500:restricted online version (Google Books)
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490:. Dover Publications 1965, p. 359-360.
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512:. Princeton University Press 2004,
457:is the orthic triangle of triangle
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311:Hooke's Law
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548:, German)
453:Triangle
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283:altitudes
267:altitudes
255:perimeter
245:in 1775:
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468:See also
287:calculus
273:Solution
231:geometry
566:in the
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