In the Airport Problem activity from Rethinking Proof, it was assumed that the 3 cities were more or less of equal size. If the cities are of different size, one can experimentally solve the problem as shown in the 'Airport 4' sketch (in the Airport Problem activity) by 'weighting' the distances proportionally to the sizes of the cities. However, the problem of cities of different sizes can also be solved purely geometrically as elegantly shown below by the Polish mathematician Hugo Steinhaus (1951).
"No science strengthens faith in the power of the human mind as much as mathematics." - Hugo Steinhaus (1887–1972)
Weighted Airport Problem
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?
Steinhaus 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. (Note: The concurrent circumcircles follow from the result expressed and illustrated by the 'Similar Triangles' sketch in the Fermat-Torricelli Point activity - click on the 'Link to similar triangles' button).
Investigate
Drag point P in the dynami sketch below 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.
Weighted Airport Problem
Challenge
Can you explain why (prove that) the above construction provides a geometric solution to the '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.
Related Links
Airport Problem (Rethinking Proof activity)
Fermat-Torricelli Point (Rethinking Proof activity)
Fermat Torricelli Similar Polygons Concurrency
Distances in an Equilateral Triangle (Viviani's theorem, Rethinking Proof activity)
Napoleon's Theorem (Rethinking Proof activity)
Napoleon's Theorem: Generalizations, Variations & Converses
Some Triangle Generalizations of Napoleon's Theorem
Miquel's Theorem (Rethinking Proof activity)
Minimum Area of Miquel Circle Centres Triangle
A variation of Miquel's theorem and its generalization
Fermat-Torricelli Point Generalization (Jacobi's theorem) plus Further Generalizations
Kosnita's Theorem
Dual to Kosnita (so-called De Villiers Points of a Triangle)
Another concurrency related to the Fermat point of a triangle
Anghel's Hexagon Concurrency theorem (Click on the 'Link to ...' button)
Some Variations of Vecten configurations
Some Circle Concurrency Theorems
External Links
Hugo Steinhaus (Wikipedia)
Fermat point (Wikipedia)
Fermat Points (Wolfram MathWorld)
The Fermat Point and Generalizations (Cut The Knot)
Steiner tree problem (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
********************************
Back to "Dynamic Geometry Sketches"
Back to "Student Explorations"
Created by Michael de Villiers, 25 July 2010, modified 21 January 2019; updated 28 November 2020; 23 Feb 2026.