Geometrical solution of weighted Fermat problem about triangles

Abstract

The paper considers weighted generalization of a classic Fermat problem about triangles: given A, B, C vertices of a triangle and three real positive numbers $\bar a,\bar b,\bar c,$ find in plane a point P minimizing the sum of its distances to A, B, C, multiplied by $\bar a,\bar b,\bar c,$ respectively. It is shown by simple geometrical methods that, if $\bar a,\bar b,\bar c,$ satisfy the triangle inequalities and further conditions also involving the angles of the triangle ABC, then there is one and only one minimum interior point of the triangle and a construction is supplied which enables us to find the minimum point by “ruler and compasses”. In all other cases, one identified vertex of the triangle is the minimum point.