A journey through the wonders of Fermat's Point - ISISS M. Casagrande (Pieve di Soligo)

The Fermat point is a remarkable point of a triangle introduced by the French mathematician Pierre de Fermat in the XVII century. In this article we studied some known properties of such point, also managing to discover some new interesting characteristics. We started analysing triangles, moving to quadrilaterals and finally to polygons. Regarding triangles, by slightly modifying the definition of Fermat point, we obtained a new set of points and we proved that the Fermat point belongs to this set as well as the triangle centroid and the orthocenter. Moving away from triangles, we obtained some results involving regular polygons, but because of the complexity of the task we were not able to find a method to determine the Fermat point in irregular polygons. For this reason, we used calculus to develop an algorithm to approximate Fermat point.