Airport Problem

"Nothing takes place in the world whose meaning is not that of some maximum or minimum." — Leonhard Euler (1707–1783)

The dynamic geometry activities below are from the "Proof as Challenge" section of my book Rethinking Proof (free to download).

Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Airport Worksheet & Teacher Notes.

Prerequisites
i) Students should have some familiarity with rigid transformations and their properties, particularly rotations.
ii) Though this activity can be done independently of the preceding activity, it is recommended that students first complete The Fermat-Torricelli Point activity before engaging with this one.

Airport Problem
Suppose an airport is planned to service three cities of approximately equal size. The planners decide to locate the airport so that the sum of the road distances to the three cities is a minimum. Where should the airport be located? Use the dynamic sketch below to explore this problem.

Airport Problem

Notes
1) Users are strongly encouraged to download the accompanying Worksheet, and to print it out to use in conjunction with the dynamic sketch above. Alternatively, copy the URL: http://dynamicmathematicslearning.com/airport-problem-rethinking-proof.pdf and paste it into a new browser window. Then resize the new window to place it side by side with this one, or one below the other. (However, this is not likely to be a feasible option for users using small screens such as a cellphone or tablet.)
Modelling Assumptions
2) The dynamic sketch of the three cities above is an example of a mathematical model that can be used to represent and analyze real-world situations. However, real-world situations are extremely complex and usually have to be simplified before mathematics can be meaningfully applied to them. What are some of the assumptions that were probably made to simplify the original problem?

Investigate & Conjecture
3) Drag point D until the sum of the distances to the three cities is a minimum. Search patiently and logically.
4) What are the measures of angles ADC, BDA, and CDB when the sum of the distances is a minimum?
5) Click on 'Show angles measures at D' button
. What do you notice about these three angles when the sum of the distances is a minimum?
6) Drag A, B, or C to a different position, but make sure △ABC remains acute. Again, drag D to obtain the optimal point for this new triangle.
7) Compare the new measurements of angles ADC, BDA, and CDB with those in Question 4. What do you notice?
8) Use your observations to write a conjecture.

Challenge
9) How certain are you that your conjecture in Q7 is correct?
10) Can you prove that your conjecture is correct?
Proof Hint
11) Click on 'Show and rotate triangle ADC' button.
12) Now work through the questions in the guided proof in the accompanying Worksheet to develop a proof.

Looking Back
You may have noticed earlier that the optimal point for the airport is the Fermat-Torricelli Point, discussed in the preceding activity. Show that the construction in this activity is equivalent to constructing equilateral triangles A'AC, B'BA, and C'CB on the sides of △ABC (and constructing lines A'B, B'C, and C'A to meet at D.

Historical Notes
Pierre de Fermat (1601 – 1665) appears to have first posed the 'airport problem' in an essay on optimization. He wanted to find a point inside an acute triangle such that the sum of the distances to the three vertices is a minimum.
.....The Italian mathematician and scientist Evangelista Torricelli (1608 – 1647) proposed constructing equilateral triangles on the sides of any triangle to locate the optimal point (Monks, 2021; also see the preceding activity The Fermat-Torricelli Point). Although this solution was proposed in 1640, it was published in 1659 by Viviani, one of Torricelli’s students.
.....The solution & guided proof described in this activity was more recently invented in 1929 by the German mathematician Joseph Hoffman (1900–1973) - see Spindler (2025). Several other famous mathematicians — for example, Gauss and Steiner — have investigated the problem and have produced some interesting generalizations.

Further Exploration
13) Can you relate the airport problem to the result discovered and proven in the activity Distances in an Equilateral Triangle (Viviani's theorem), and use it to develop an indirect proof for the optimal placement of the airport? (Click on 'Link to Airport 2' button and use the new sketch to investigate the relationship.)
14) Where should the airport be placed if the cities lie in the shape of an obtuse triangle with one of the angles greater than 120°? (Drag your original triangle into an obtuse triangle and investigate). Can you prove that your solution is optimal in this obtuse case?
15) Where should the airport be placed if the three cities all lie in a straight line (are collinear)? Can you generalize? (Click on 'Link to Airport 3' button and investigate). Can you prove that your solutions are optimal in these odd and even collinear cases?
(Note: A modified version of the collinear problem above was asked in the 2nd second round South African Mathematics Olympiad paper in 1998 - see SA Mathematics Olympiad Problem 1998 R2 Q18).
16) Where should the airport be placed if the cities are all of different sizes, for example, if A, B, and C respectively have 60,000, 100,000, and 70,000 people? (Click on 'Link to Airport 4' button and investigate). For a purely geometric solution go to Weighted Airport Problem.
17) Where should the airport be placed if there are four cities instead of only three? (Click on 'Link to Airport 5' button and investigate). Is your solution also valid if the four cities lie in the shape of a concave quadrilateral? Can you prove that your solution is correct?
18) Where should a spaceport be built for four planets that lie in the shape of a tetrahedron so that the total sum of distances from the spaceport to the planets is a minimum? (Compare with Hanson, 2016).

Additional Properties of the Configuration
The configuration of a triangle with equilateral or other triangles constructed on its sides is geometrically very rich. In the following web activity, namely, 'Napoleon's theorem' (not yet active), we will further explore some more properties of this configuration.

Soap Bubble Geometry & Steiner Networks
A beautiful way to experimentally illustrate the Fermat-Torricelli point is by using Soap Bubble Geometry. The physical properties of soap film are such that they would always tend to a shape or position of 'minimal energy'. For example the first figure below (from Park & Flores, 2014), show a soap bubble solution for the classic Fermat problem - the 120° angles clearly visible.
.....Soap Bubble Geometry can therefore also be used to model solutions for the shortest road networks between 3, 4, 5, etc. cities (so-called Steiner networks). The second figure below shows an example of a physical soap bubble demonstration of a minimal path (shortest road network) for a square. Demonstrations such as these were sometimes used by me in lectures at UKZN (University of KwaZulu-Natal) and at some AMESA (Association for Mathematics Education of South Africa) Conferences.
.....Recommended Video: Watch an excellent YouTube Video by Frank Morgan on Soap Bubbles and Mathematics.

soap bubble minimal path fermat ......................... soap bubble minimal path square

The Fermat-Torricelli Point in other Geometries
Another interesting, educationally productive exploration for students to investigate is Fermat's original problem of finding a point P inside a scalene triangle that minimizes the sum of the distances from P to the vertices on other surfaces or geometries. For example, using the well-known Lénárt Sphere, they can experimentally investigate the problem on the surface of a sphere - see for example Lénárt (2005). Some readers might also be interested in using Cinderella, which is dynamic geometry software (freeware) that one can use to explore problems dynamically, not only in Euclidean geometry, but also in spherical and hyperbolic geometry. In addition, ambitious students might even want to explore the problem more generally by looking at corresponding (weighted) solutions for elliptic and hyperbolic geometry - compare for example, Zachos & Zouzoulas (2008).
.....As also mentioned in several of my talks and workshops on the 'Airport Problem' (Fermat-Torricelli point) at NCTM and AMESA conferences over the years (going back to at least about 1999), investigating the Fermat-Torricelli point in a taxicab geometry environment using dynamic geometry is also a fun, productive and relatively easy mathematical exercise for students to engage with - see for example the Building a Bus Stop activity and/or Hanson (2015).

References
De Villiers, M. (1995). A Generalisation of the Fermat-Torricelli Point. The Mathematical Gazette, 9(485) (July), pp. 374-378.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Hanson, J.R. (2015). The Fermat point for a taxicab triangle. International Journal of Mathematical Education in Science and Technology, April 201546(3), DOI: 10.1080/0020739X.2016.1259515.
Hanson, J.R. (2016). Fermat point for a triangle in three dimensions using the taxicab metric. International Journal of Mathematical Education in Science and Technology, December 201648(3):1-10, DOI: 10.1080/0020739X.2016.1259515.
Lénárt, I. (2005). The Fermat-Torricelli Point in Three Geometries. (Based on Szmerka, G. (2005) The Fermat-Torricelli problem in plane and spherical geometry from educational, mathematical and historical standpoints. Thesis for Faculty of Natural Sciences, Eötvös Loránd University, Budapest (in Hungarian). Supervisor: I. Lénárt).
Monks, K. M. (2021). The Fermat-Torricelli Point and Cauchy's Method of Gradient Descent. Calculus. 19. https://digitalcommons.ursinus.edu/triumphs_calculus/19/.
Park, J. & Flores, A. (2014). Fermat's point from five perspectives. International Journal of Mathematical Education in Science and Technology, DOI: 10.1080/0020739X.2014.979894.
Spindler, K. (2025). The Fermat-Torricelli problem revisited. Mathematische Semesterberichte, 1 Dec 2025, DOI: https://doi.org/10.1007/s00591-025-00402-y.
Zachos, A. N. & Zouzoulas, G. (2008). The Weighted Fermat-Torricelli Problem and an "Inverse" Problem. Journal of Convex Analysis, Vol. 15, No. 1, pp. 55–62.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
Rigid Transformations (MathBitsNotebook)
Rotation (mathematics) (Wikipedia)
Pierre de Fermat (Wikipedia)
Evangelista Torricelli (Wikipedia)
Fermat point (Wikipedia)
Fermat Points (Wolfram MathWorld)
The Fermat Point and Generalizations (Cut The Knot)
Steiner tree problem (Wikipedia)
Soap Bubbles and Mathematics (YouTube video)
Taxicab geometry (Wikipedia)
Taxicab Geometry (pdf, UCI Math Circle)
Cinderella (Free Interactive Geometry Software)
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.)

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Michael de Villiers, created with WebSketchpad, 21 Feb 2026; updated 25 Feb 2026; 2 March 2026; 21 April 2026.