Properties of Fermat Point

There are various properties associated with Fermat Point, some of these properties are:

• If the triangle is equilateral, then the Fermat point will coincide with one of the vertex. Because in equilateral distance between any point within it is always equal.
• In a triangle whose all three interior angles are less than 120°, the Fermat point lies inside a triangle.
• When the largest angle of the triangle is not larger than 120°, the first isogonic center is the Fermat point.
• It lies on the circumcircle of the triangle.
• It is unique or we can say that in a triangle, there will be one and only one Fermat point exists.
• If We make a reflection of any of the triangle’s vertices across the corresponding side then the reflected vertex will lie on the line connecting the other two vertices. In other words, we can say that the reflected vertex will be collinear with the original triangle’s vertices and the Fermat point.
• And If one of the interior angles is greater than 120°, then the Fermat point lies outside a triangle.
• The circumcircles of the three constructed equilateral triangles are concurrent at Fermat Point.
• Trilinear coordinates for the first isogonic center:

[Tex]\displaystyle {\begin{aligned}&\csc \left(A+{\tfrac {\pi }{3}}\right):\csc \left(B+{\tfrac {\pi }{3}}\right):\csc \left(C+{\tfrac {\pi }{3}}\right)\\&=\sec \left(A-{\tfrac {\pi }{6}}\right):\sec \left(B-{\tfrac {\pi }{6}}\right):\sec \left(C-{\tfrac {\pi }{6}}\right).\end{aligned}} [/Tex]

• Trilinear coordinates for the second isogonic center:

[Tex]\displaystyle {\begin{aligned}&\csc \left(A-{\tfrac {\pi }{3}}\right):\csc \left(B-{\tfrac {\pi }{3}}\right):\csc \left(C-{\tfrac {\pi }{3}}\right)\\&=\sec \left(A+{\tfrac {\pi }{6}}\right):\sec \left(B+{\tfrac {\pi }{6}}\right):\sec \left(C+{\tfrac {\pi }{6}}\right).\end{aligned}} [/Tex]

• Trilinear coordinates for the Fermat point:

[Tex]\displaystyle 1-u+uvw\sec \left(A-{\tfrac {\pi }{6}}\right):1-v+uvw\sec \left(B-{\tfrac {\pi }{6}}\right):1-w+uvw\sec \left(C-{\tfrac {\pi }{6}}\right) [/Tex]

where u, v, and w respectively denote the Boolean variables (A < 120°), (B < 120°), and (C < 120°).

• The isogonal conjugate of the first isogonic center is the first isodynamic point:

[Tex]\displaystyle \sin \left(A+{\tfrac {\pi }{3}}\right):\sin \left(B+{\tfrac {\pi }{3}}\right):\sin \left(C+{\tfrac {\pi }{3}}\right). [/Tex]

• The isogonal conjugate of the second isogonic center is the second isodynamic point:

[Tex]\displaystyle \sin \left(A-{\tfrac {\pi }{3}}\right):\sin \left(B-{\tfrac {\pi }{3}}\right):\sin \left(C-{\tfrac {\pi }{3}}\right). [/Tex]

• The lines joining the first isogonic center and first isodynamic point, and the second isogonic center and second isodynamic point are parallel to the Euler line. The three lines meet at the Euler infinity point.

These properties highlight the geometrical significance of the Fermat Point and its interesting characteristics of it, making it an interesting concept in the field of geometry.

Fermat Point is the point that Pierre de Fermat, the 17th-century French mathematician, posed as a challenge to his compatriot Evangelista Torricelli to geometrically determine, is named in Fermat’s honour as the solution that would minimize the total combined distance from each vertex of a triangular figure to any single internal locus. Torricelli solved the problem, therefore other than Fermat Point, it is also known as the Fermat Point or the Torricelli Point or Fermat Torricelli Point.

Although various methods exist to locate the Fermat point, connecting the vertices of the original triangle to those of the equilateral triangles constructed on each of its sides furnishes a straightforward technique. The intersection of these segments is the Fermat point.

The Fermat point gives a solution to the geometric median and Steiner tree problems for three points.