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.

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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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Contributed by: E. Coiras (July 2020)
Open content licensed under CC BY-NC-SA


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If all the vertex angles are less than 120, ; 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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