The dynamic geometry activities below are from the "Proof as Explanation" 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: Water Supply: Four Towns Worksheet & Teacher Notes.
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In a developing country such as South Africa, there are many remote rural areas where people do not have access to safe, clean water and are dependent on nearby rivers and streams for their water supply. Apart from being unreliable due to frequent droughts, these rivers and streams are often muddy and unfit for human consumption. Suppose the government wants to build a water reservoir and purification plant for four villages in such a remote rural area. Where should the government place the water reservoir so that it is the same distance from all four villages?
Water Supply: Four Towns
Notes
1) The main purpose of this activity is to introduce students to the following concepts: equi-distance, perpendicular bisector, concurrency, cyclic quadrilateral.
2) To measure the distances from P to the four villages as directed in the worksheet, use the toolbar on the right of the webpage.
3) To construct the perpendicular bisectors as directed in the worksheet, use the vertical slider just to the right of the toolbar to scroll down & select the 'Perp. Bisector' tool.
4) To use the Compass tool, scroll down in the toolbar to select it.
5) As asked in Q2 of the Further Exploration in the accompanying worksheet, make a list of all the possible simplifying assumptions of the real world that have been made in the above problem. Check your answer with the accompanying Teachers Notes at the link given at the top. Read more about an empirical study by Mudaly & De Villiers (2004) and Mudaly (2007) related to modeling this particular problem and how it helped provide provided a meaningful context to develop a need for proof.
6) To investigate the location of the best position to place the water reservoir for four towns when there is no equi-distant point, click on the 'Link to Water Supply: Finding Minimum' button to navigate to a new sketch.
7) In a classroom discussion in 2006 at Kennesaw State University, an undergraduate student, Renate Lebleu Davis, proposed the intersection of the diagonals of the quadrilateral formed by the perpendicular bisectors of the (non-cyclic) quadrilateral in 6) as a possible solution. This subsequently led to the discovery and proof that this intersection point of its diagonals was equi-distant from both pairs of opposite vertices of the original quadrilateral - for more information about this go to this link: The quasi-circumcentre of a quadrilateral.
8) It is recommended to complete this activity for four towns before the Water Supply: Three Towns activity.
References
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2025). Another Student Discovery: The Quasi-Circumcentre and Quasi-Incentre of a Quadrilateral. Learning and Teaching Mathematics, No. 39 (Dec), pp. 31-35.
Mudaly, V. & De Villiers, M. (2004). Mathematical Modeling and Proof. Paper presented at 10th AMESA Congress, 30 June -4 July 2004, University of the North-West, Potchefstroom.
Mudaly, V. (2007). Can our Learners Model in Mathematics?. The Mathematics Enthusiast, Vol. 4, No. 1, Article 7.
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Related Links
Water Supply: Three Towns (Rethinking Proof activity)
Three Circle Geometry Theorem Proofs by Transformations
The Perpendicular Bisectors of a Parallelogram
The Perpendicular Bisectors of an Apollonius Quadrilateral
A theorem involving the perpendicular bisectors of a hexagon with opposite sides parallel
The Equi-inclined Bisectors of a Cyclic Quadrilateral
Perpendicular Bisectors of Circumscribed/Tangential Quadrilateral Theorem
Triangle Altitudes (Rethinking Proof activity)
Generalizing the concept of perpendicular bisectors to 3D
Visually Introducing & Classifying Quadrilaterals by Dragging (Grades 1-7)
Jannie's Father's Farm Problem (mathematical modelling)
Angle Divider Theorem for a Cyclic Quadrilateral
Distances in an Equilateral Triangle (Rethinking Proof activity)
Some Trapezoid (Trapezium) Explorations (See Investigation 5)
The quasi-circumcentre of a quadrilateral
Two British Mathematics Olympiad Concurrency Problems
Carnot's (or Bottema's) Perpendicularity Theorem & Some Generalizations
A generalization of Neuberg's Theorem & the Simson line
External Links
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, 20 June 2025; updated 24 June 2025; 8 Jan 2026; 23 Feb 2026.