Suppose one wants to build an airport for 3 cities with populations in the ratios 5:7:9 so that the sum of the 'weighted' distances from the airport to the 3 cities is a minimum. Construct a triangle with the three sides in the same ratios 5:7:9 and measure its angles, a, b and c as indicated. Then construct on the sides of the red triangle, determined by the locations of the cities, the triangles as shown by using the measured angles a, b and c. Then the generalized Fermat-Torricelli point as indicated by the lines of concurrency (as well as the concurrent circumcircles) is the desired location for the airport.
Drag point P and observe how the sum of the weighted distances x, y and z from P to the vertices changes, and convince yourself that the Fermat-Torricelli point is where this sum is minimized.
(Note: The concurrent circumcircles follow from the result expressed and illustrated by the 2nd generalization of Napoleon's theorem in the sketch Napoleon Generalization 2)
Weighted Airport Problem
Created by Michael de Villiers, 25 July 2010.