Construction
In order to locate the Fermat point of a triangle with largest angle at most 120°:
- Construct two equilateral triangles on any of the three sides of the given triangle.
- For each new vertex of the equilateral triangle, draw a line from it to the opposite triangle's vertex.
- The two lines intersect at the Fermat point.
An alternate method is the following:
- 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.
- 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.
- 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.
Read more about this topic: Fermat Point
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