An algebraic approach to finding the Fermat–Torricelli point

Abstract

Using a calculus and an algebraic approach, the Cartesian coordinates of the Fermat–Torricelli point are deduced for triangles with no internal angle greater than 120°. Although in theory, the deduction of these coordinates could be made ‘by hand’, in practice it is very laborious to obtain them without the aid of mathematical computer software, but with human guidance, since there are mathematical artifices not yet incorporated into the software. It is also shown that these coordinates can be conveniently expressed in terms of the side lengths and the area of the triangle. These coordinates are contrasted with the coordinates of a similar point: one whose sum of the squares of the distances to the vertices of an arbitrary triangle is a minimum.

Keywords:

• fermat point
• minimal sum of distances
• algebraic solution

Subject classification codes:

• 14Q99

Disclosure statement

No potential conflict of interest was reported by the authors.