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.