In a given acute triangle 
, find the inscribed triangle whose perimeter
is as small as possible. The answer is the orthic
triangle of 
.
The problem was proposed and solved using calculus by Fagnano in 1775 (Coxeter and
Greitzer 1967, p. 88).
Fagnano's Problem
See also
Acute Triangle, Orthic Triangle, PerimeterExplore with Wolfram|Alpha
References
Bogomolny, A. "Fagnano's Problem: What Is It?" http://www.cut-the-knot.org/Curriculum/Geometry/Fagnano.shtml.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 21, 1969.Coxeter, H. S. M. and Greitzer, S. L. "Fagnano's Problem." §4.5 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 88-89, 1967.Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 347, 1996.Ha, N. M. "Another Proof of Fagnano's Inequality." Forum Geom. 4, 199-201, 2004. http://forumgeom.fau.edu/FG2004volume4/FG200422index.html.Kazarinoff, N. D. Geometric Inequalities. New York: Random House, pp. 76-77, 1961.Morley, F. and Morley, F. V. Inversive Geometry. Boston, MA: Ginn, p. 37, 1933.Referenced on Wolfram|Alpha
Fagnano's ProblemCite this as:
Weisstein, Eric W. "Fagnano's Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FagnanosProblem.html