Fermat Point - Construction

Construction

In order to locate the Fermat point of a triangle with largest angle at most 120°:

1. Construct two equilateral triangles on any of the three sides of the given triangle.
2. For each new vertex of the equilateral 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 lying outside the original triangle.
2. Draw two circles, each with a 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. Their relationship is explored below.