What is Fermat Point or Torricelli Point?

This point is related to the concept of Geometry. It is the point in inside a triangle from which the sum of the distance between vertices is minimum. the Fermat point can be found by constructing an equilateral triangle on the sides of the original triangle.

It is also known as the Torricelli point or the Fermat–Torricelli point. It is named after Pierre de Fermat and Evangelista Torricelli, who independently solved or worked on the problem of finding the Fermat point in the 17th century.

This question is proposed by Fermat, as a challenge in front of Italian mathematician Evangelista Torricelli. And He solved the problem in a similar way to Fermat, but the difference is that he used a circumcircle instead of three equilateral triangles as solved by Fermat.

Fermat Point Definition

In a Triangle, the sum of the distances to the three vertices is smallest from point which is known as Fermat Point. It is also known as Torricelli point or Fermat Torricelli Point. It is named after a French Mathematician, Pierre de Fermat. And it is named as Torricelli Point after a name of Italian Mathematician, Evangelista Torricelli.

Terminologies related to Fermat Point

Some of the common terms used to explain Fermat point and its construction are listed below:

• First Isogonic center: The first isogonic center of a triangle is a point in the plane of the triangle at which the three sides subtend each an angle of 120°.
• Second Isogonic center: The second isogonic center is that point at which two sides subtend each at an angle of 60° and the third side an angle of 120°.
• First Isodynamic center: The first isodynamic point is the intersection of three circles through the pairs of points AB, AC, and BC, where each of these circles intersects the circumcircle of △ABC to form a lens with an apex angle of 2π/3.
• Second Isodynamic Point: The Second Isodynamic Point is the intersection of three circles that intersect the circumcircle to form lenses with an apex angle of π/3.
• Euler Line: The Euler line of a triangle is a line going through several important triangle centers, including the orthocenter, circumcenter, centroid, and center of the nine-point circle.
• Euler Infinity Point: The Euler infinity point is the intersection of the Euler line and line at infinity. Since it lies on the line at infinity, it is a point at infinity.

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.