# Geometrical solution of weighted Fermat problem about triangles

• Mario Martelli
Chapter
Part of the Applied Optimization book series (APOP, volume 13)

## 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.

## Keywords

Triangle Inequality Real Positive Number Minimum Point Orthogonal Axis Weber Problem
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