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: Three Towns Worksheet & Teacher Notes.
Prerequisite
Though not essential, it is highly recommended to first complete the activity for four towns at Water Supply: Four Towns before completing the activity below.
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The government needs to build a water reservoir for the three villages shown below. Where should the water reservoir be placed so that it is equidistant from all three villages?
Water Supply: Three Towns
Notes
1) The main purpose of this activity is to introduce students to the concurrency of the perpendicular bisectors of a triangle and its logical explanation (proof) in terms of equi-distance.
2) To construct the perpendicular bisectors as directed in the worksheet, select & use the 'Perp. Bisector' tool on the left.
3) To investigate Q2 in the Further Exploration as directed in the worksheet, click on the 'Link to Further Exploration' button to navigate to a ready-made sketch.
4) Can you explain why (prove that) your generalization in 3) is true?
5) Check your answer to 5) by clicking on this 2006 Classroom Note from Kennesaw State University.
6) In a classroom discussion with under-graduate students in 2011, a student pointed out that when the villages are positioned in the shape of an obtuse triangle, then for practical, real world purposes, the equi-distant point (circumcentre) might not be the 'best' choice for locating the water reservoir. This led to an interesting class discussion about 'mathematical modelling' and its relation to the 'real world' as reported in De Villiers (2013).
References
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2013). Equality is not always 'best'!. Learning & Teaching Mathematics, No. 14, pp. 17-21.
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Related Links
Water Supply: Four Towns (Rethinking Proof activity
Triangle Altitudes (Rethinking Proof activity)
Generalizing the concept of perpendicular bisectors to 3D
Visually Introducing & Classifying Quadrilaterals by Dragging (Grades 1-7)
Distances in an Equilateral Triangle
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
Power Lines of a Triangle
A theorem involving the perpendicular bisectors of a hexagon with opposite sides parallel
Perpendicular Bisectors of Circumscribed/Tangential Quadrilateral Theorem
The Perpendicular Bisectors of an Apollonius Quadrilateral
The Equi-inclined Bisectors of a Cyclic Quadrilateral
Angle Divider Theorem for a Cyclic Quadrilateral
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, 22 June 2025; updated 24 June 2025; 6 Oct 2025.