Fagnano problem

Altitude triangle: inscribed triangles: ${\displaystyle \triangle DEF}$
${\displaystyle \triangle DEF\,,\triangle GHI}$
${\displaystyle |DE|+|EF|+|FD|\leq |GH|+|HI|+|IG|}$

The Fagnano problem is the following optimization problem named after Giovanni Fagnano .

Find the triangle of minimal circumference inscribed in an acute triangle .

Here, an inscribed triangle of a triangle is understood to mean a triangle whose corners are on the sides of the triangle , that is , and . It applies to the height base triangle that its circumference is smaller than that of any other inscribed triangle and thus it is the solution to the Fagnano problem. ${\displaystyle \triangle ABC}$ ${\displaystyle \triangle GHI}$ ${\displaystyle \triangle ABC}$ ${\displaystyle G\in {\overline {AC}}}$ ${\displaystyle H\in {\overline {AB}}}$ ${\displaystyle I\in {\overline {BC}}}$ ${\displaystyle \triangle DEF}$ ${\displaystyle \triangle GHI}$

First, Giovanni Fagnano's father Giulio Carlo Fagnano showed that for any fixed point U on 2 points V up and W up can be constructed in such a way that the circumference of the triangle is minimal. Giovanni Fagnano used this result in order to determine with the help of the differential calculus from all possible U on that for which the circumference of the triangle is minimal. Several elementary geometrical proofs were later found, including by Leopold Fejér and Hermann Amandus Schwarz . These proofs mostly use properties of reflections to determine a minimal path. ${\displaystyle {\overline {AB}}}$ ${\displaystyle {\overline {AC}}}$ ${\displaystyle {\overline {BC}}}$ ${\displaystyle \triangle UVW}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle \triangle UVW}$

literature

• Heinrich Dörrie : triumph of mathematics. 100 famous problems from 2 millennia of mathematical culture. 2nd, supplemented edition. F. Hirt, Breslau 1940, problem 90.
• 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, Princeton NJ u. a. 2004, ISBN 0-691-07078-4 , p. 67.
• HSM Coxeter , Samuel L. Greitzer: Timeless Geometry. Klett, Stuttgart 1983, ISBN 3-12-983390-0 .
• Hermann Amandus Schwarz : Collected Mathematical Treatises . Volume 2. Berlin 1890, pp. 344-345 Textarchiv - Internet Archive

Web links

• Wolfgang Zimmer: An open task on the subject of "Minimal distances" (Sinus materials) (PDF; 141 kB)
• Fagnano problem on cut-the-knot
• Fagnano's problem in the Encyclopedia of Mathematics (English)
• Fagnano problem on triangle geometry website
• Eric W. Weisstein : Fagnano's problem . In: MathWorld (English).