First Fermat Point and Isogonic Center of a Triangle

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.
Let be the sum of the distances from a point to the three vertices of . Then minimizes . The contours show the level lines of .



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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.
[1] Wikipedia. "Fermat Point." (Jul 13, 2020)
[2] C. Kimberling. "X(13)." Encyclopedia of Triangle Centers. (Jul 13, 2020)
[3] Wikipedia. "Geometric Median." (Jul 13, 2020)
[4] Wikipedia. "Barycentric Coordinates." (Jul 13, 2020)
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