In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 the Fermat point of a triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

, also called Torricelli point, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible. It is so named because this problem is first raised by Fermat

Pierre de Fermat

Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...

 in a private letter.

The Fermat point gives a solution to the Steiner tree problem for three points.
In order to locate the Fermat point of a triangle with largest angle at most 120°:

1. Construct two regular triangles out of any the three sides of the given triangle.
2. For each new vertex
   Vertex (geometry)
   In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
   
    of the regular triangle, draw a line from it to the opposite triangle's vertex.
3. The two lines intersect at the Fermat point.

An alternate method is the following:

1. On any two of the three sides, construct two isosceles triangles, with base the side in question, 30-degree angles at the base, and vertices laying outside the original triangle.
2. Draw two circles, each with center on the vertex of the just constructed isosceles triangles and radius the identical side of the isosceles triangles.
3. The intersection inside the original triangle between the two circles is the Fermat point.

When a triangle has an angle greater than 120° the Fermat point is sited at the obtuse angled vertex.

The Fermat point has a near-identical twin called the first isogonic center or X(13) and it is important not to confuse the two.