The first Fermat point 
(or 
) (sometimes simply called "the Fermat point,"
Torricelli point, or first isogonic center) is the point 
which minimizes the sum of distances from 
, 
, and 
in an acute triangle,
 |
(1) |
It has equivalent triangle center functions
and is Kimberling center 
(Kimberling 1998, p. 67).
It also arises in Napoleon's theorem.
The antipedal triangle of the first Fermat point is an equilateral triangle (Shenghui Yang, pers. comm. to E. Pegg, Jr., Jan. 3, 2025).
See also
Fermat Axis, Fermat Points, Napoleon's Theorem, Second Fermat Point
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References
Kazarinoff, N. D. Geometric Inequalities. New York: Random House, pp. 117-118, 1961.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Fermat Point." http://faculty.evansville.edu/ck6/tcenters/class/fermat.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(13)=1st Isogonic Center." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X13.
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Cite this as:
Weisstein, Eric W. "First Fermat Point." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FirstFermatPoint.html