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.[more]
Let be the sum of the distances from a point to the three vertices of . Then minimizes . The contours show the level lines of .[less]
Contributed by: E. Coiras (July 2020)
Open content licensed under CC BY-NC-SA
If all the vertex angles are less than 120º, ; otherwise is the obtuse-angled vertex . The use of a closed formula removes the need for such conditional checks. is also the 2D geometric median  of the triangle vertices, which as shown here can be calculated by computing two scalar (1D) medians on the corresponding barycentric coordinates . A closed formula for the median of three scalars is also introduced to remove implicit conditional checks.
 Wikipedia. "Fermat Point." (Jul 13, 2020) en.wikipedia.org/wiki/Fermat_point.
 C. Kimberling. "X(13)." Encyclopedia of Triangle Centers. (Jul 13, 2020) faculty.evansville.edu/ck6/encyclopedia/ETC.html.
 Wikipedia. "Geometric Median." (Jul 13, 2020) en.wikipedia.org/wiki/Geometric_median.
 Wikipedia. "Barycentric Coordinates." (Jul 13, 2020) en.wikipedia.org/wiki/Barycentric_coordinate_system.