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. Where should the airport be built to achieve this?
Solution: 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
Reference: Steinhaus, H. (1951). Mathematical Snapshots, Oxford University Press, New York.
Further readings
Gueron, S. & Tessler, R. (2002). The Fermat-Steiner Problem. The American Mathematical Monthly, Vol. 109, No. 5 (May), pp. 443-451.
Shen, Y. & Tolosa, J. (2008). The Weighted Fermat Triangle Problem. International Journal of Mathematics and Mathematical Sciences, March 2008, Article ID 283846,16 pp.
Created by Michael de Villiers, 25 July 2010, modified 21 January 2019; 28 November 2020.