First Fermat Point and Isogonic Center of a Triangle
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This Demonstration plots the first Fermat point 
(red) and first isogonic center 
(yellow) of a triangle 
. You can drag the triangle vertices. When 
and 
are both inside the triangle, they coincide.
Contributed by: E. Coiras (July 2020)
Open content licensed under CC BY-NC-SA
Snapshots
Details
If all the vertex angles are less than 120º, 
; otherwise 
is the obtuse-angled vertex [2]. The use of a closed formula removes the need for such conditional checks. 
is also the 2D geometric median [3] of the triangle vertices, which as shown here can be calculated by computing two scalar (1D) medians on the corresponding barycentric coordinates [4]. A closed formula for the median of three scalars is also introduced to remove implicit conditional checks.
References
[1] Wikipedia. "Fermat Point." (Jul 13, 2020) en.wikipedia.org/wiki/Fermat_point.
[2] C. Kimberling. "X(13)." Encyclopedia of Triangle Centers. (Jul 13, 2020) faculty.evansville.edu/ck6/encyclopedia/ETC.html.
[3] Wikipedia. "Geometric Median." (Jul 13, 2020) en.wikipedia.org/wiki/Geometric_median.
[4] Wikipedia. "Barycentric Coordinates." (Jul 13, 2020) en.wikipedia.org/wiki/Barycentric_coordinate_system.
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