The first Fermat point acute triangle ,
(1)
It has equivalent triangle center functions
and is Kimberling center
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. 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 . Referenced
on Wolfram|Alpha First Fermat Point
Cite this as:
Weisstein, Eric W. "First Fermat Point."
From MathWorld https://mathworld.wolfram.com/FirstFermatPoint.html
Subject classifications
(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,
![bc[c^2a^2+(c^2+a^2-b^2)^2][a^2b^2-(a^2+b^2-c^2)^2][4Delta-sqrt(3)(b^2+c^2-a^2)]](/images/equations/FirstFermatPoint/Inline12.svg)
(Kimberling 1998, p. 67).
