Fermat Point

Sun Jun 21 2015

In geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat-Torricelli point, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible.

Let’s say we have a triangle with vertices A, B & C. If the largest angle of this triangle is ≤ 120°, then the Fermat point is the same as the first isogonic center. We can calculate the isogonic center by:

  1. Constructing an equilateral triangle along each side of the main triangle.
  2. Constructing these equilateral triangles introduces 3 new vertices: P, Q & R.
  3. Draw a line from each of these new vertices to the vertex — of the main triangle — directly opposite to them, for example: A to R.
  4. The point of intersection of these 3 lines in the isogonic center.

In the calculator below the point of intersection of the 3 yellow lines is the Fermat point.

However, if the triangle has an angle greater than 120° then the Fermat point is simply the vertex that is obtuse-angled. For example, if the angle at A was > 120°, then A is the Fermat point.

You can test this out using the calculator below. Click and drag the vertices to construct various types of triangles and see how the Fermat point changes.