The dynamic geometry activities below are from 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: Distances Worksheet & Teacher Notes.
Functions of Proof
This activity is intended as a first introduction to proof as a means of explanation. By further reflection on the explanatory proof of the result, students can also be introduced to the discovery function of proof, allowing them to generalize further to higher polygons.
Prerequisites
i) Area formulas for triangles, elementary algebra (factorization).
ii) Since a couple of my students appeared to have some difficulty understanding the concept of 'perpendicular distance' as the 'shortest distance' from a point to a line, I designed the following preliminary activity. Students are therefore recommended to first complete this activity: Building a Road before continuing with the activity below.
............
A shipwreck survivor manages to swim to a tropical island. As it happens, the island closely approximates the shape of an equilateral triangle. She soon discovers that the surfing is outstanding on all three of the island’s coasts. She crafts a surfboard from a fallen tree and surfs every day. Where should she build her house so that the sum of the distances from her house to all three beaches is as small as possible? (She visits each beach with equal frequency.)
'.....Before you proceed further, locate a point in the triangle at the spot where you think she should build her house.
Distances in Different Polygons
Notes
1) In modelling any real world situation, simplifying assumptions are always made. Make a list of all the possible assumptions that have been made in the above problem. Check your answer with the accompanying Teacher Notes at the link given at the top.
Investigate
2) Click on the 'Show distance sum' button, then drag P in the sketch above. What do you notice? Did you expect that or are you surprised?
Logical Explanation
3) Can you explain why the observation is true?
(Download the associated Worksheet answering questions that will guide you through a traditional logical argument and explanation in terms of the areas of triangles).
Scalene Triangle
4) To investigate where you should locate P to minimize the sum of the perpendicular distances to all three sides of a scalene triangle (as instructed in the Further Investigation of the accompanying worksheet), click on the 'Link to Distances Scalene Triangle' button to navigate to a new sketch.
5) Can you explain/prove your findings in 4)?
Directed Distances
6) For all four cases (equilateral triangle, rhombus, parallelogram, pentagon), also ensure that you drag P outside the figures.
Note: When P is dragged outside, the proofs need to be adapted using the idea of 'directed distances' - see the Teacher Notes and/or De Villiers (2020).
Converse
7) Formulate a converse for Viviani's theorem for an equilateral triangle.
8) Can you prove or disprove your formulated converse in 7)?
9) Check your answer in 8) at this link: Converse of Viviani's Theorem.
Further 2D Generalization
10) Click on the 'Link to Rhombus Distances', 'Link to Parallelogram Distances' and 'Link to Pentagon Distances' button to respectively navigate to new sketches for a rhombus, parallelogram and and equi-angled pentagon.
11) Explore each of these new figures. What do you notice? Can you explain (prove) your observations? Can you generalize further?
(Compare your answers to 10) and 11) at: 2D Generalizations of Viviani's Theorem).
3D Generalizations
12) Can you generalize Viviani's theorem and its other 2D generalizations to 3D?
(Check your answer to 10) at (some) 3D Generalizations of Viviani's Theorem).
Viviani Variation
13) For an interesting variation of Viviani's theorem, discovered by Duncan Clough, a Grade 11 student from Bishops Diocesan College, a high school in Cape Town, see De Villiers (2012).
Further 2D Generalization Hints
a) The beauty of the traditional area proof for Viviani's theorem for an equilateral triangle is that one can immediately see that it generalizes to a rhombus, and more generally to any (convex) equilateral polygon. For example, if the sides of the equilateral polygon are a, the perpendicular distances are hi and the area of the polygon is A, we immediately have as before that A = ½a∑hi, from which follows that ∑hi is constant.
b) For a proof hint for the equi-angled pentagon, click on the 'Proof Hint' button in the 'Pentagon Distances' sketch.
c) Compare your proof in b) for the equi-angled pentagon (which can be generalized to equi-angled polygons), to the one in De Villiers (2005).
Some Mathematics Education Research
14) a) Research by Mudaly & De Villiers (2000) with Grade 9 learners found that even though learners were convinced of the truth of Viviani's theorem using the dragging modality of dynamic geometry, they still exhibited a separate need for explanation which was then satisfied by means of a guided proof.
'......b) Read the research by Govender (2013) about the engagement & experiences of preservice mathematics teachers with the generalization of Viviani's theorem to equilateral polygons.
Some Applications of Viviani's Theorem
15) It can be used to prove the optimal location of the point inside an equilateral triangle which minimizes the sum of the distances from the point to its vertices - see for example, the Airport Problem activity and Solution 3 at The Fermat Point and Generalizations.
16) It can be used to solve a famous probability problem: If a stick of length x is broken into three pieces, what is the probability that the three pieces can be used to construct a triangle? See for example, The Broken Stick Problem.
Historical Background
The theorem above, about the sum of the perpendicular distances from a point to the sides of an equilateral triangle, is named after an Italian mathematician and scientist, Vincenzo Viviani (April 5, 1622 – September 22, 1703) - read more about him in the External Links below.
References & Readings
Abboud, E. (2010). On Viviani's Theorem and its Extensions. College Mathematics Journal. 43 (3), pp. 203–211.
Alsina, C. & Nelsen, R.B. (2006). Math Made Visual: Creating Images for Understanding Mathematics. The Mathematical Association of America, pp. 29-30.
Armstrong, A. & McQuillan, D. (2017). Specialization, generalization, and a new proof of Viviani's theorem. arXiv.
Chen, Z. & Liang, T. (2006). The Converse of Viviani’s Theorem. The College Mathematics Journal, 37(5), pp. 390-391.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2005). Crocodiles and Polygons. Mathematics in School, pp. 2-4.
De Villiers, M. (2012). An illustration of the explanatory and discovery functions of proof. (Clough's Theorem). Pythagoras, 33(3), 8 pages.
De Villiers, M. (2013). 3D Generalisations of Viviani's theorem. The Mathematical Gazette, Vol 97, No 540 (November), pp. 441-445.
De Villiers, M. (2020). The Value of using Signed Quantities in Geometry. Learning and Teaching Mathematics, No. 29, pp. 30-34.
Govender, R. (2013). Constructions and Justifications of a Generalization of Viviani’s Theorem. (11.8 MB). A full thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics Education), University of KwaZulu-Natal.
Jia, G. Z. et al. (2020). Viviani’s Theorem and Related Problems. Paya Lebar Methodist Girls’ School (Secondary), A project presented to the 2020 Singapore Mathematics Project Festival.
Kawasaki, K. (2005). Proof without words: Viviani's theorem. Mathematics Magazine, vol. 78, no. 3 (June), p. 213.
Kawasaki, K-I.; Yagi, Y. & Yanagawa, K. (2005). On Viviani's theorem in three dimensions. The Mathematical Gazette, Vol. 89, No. 515 (Jul), pp. 283-287.
Mudaly, V. & De Villiers, M. (2000). Learners' Needs for Conviction and Explanation within the Context of Dynamic Geometry. Pythagoras, 52, August, pp. 20-23.
Samelson, H. (2003). Proof Without Words: Viviani's Theorem with Vectors. Mathematics Magazine, Vol. 76, No. 3 (Jun.), p. 225.
Tikekar, V.G. (2015). A Proof Without Words for Viviani´s Theorem. At Right Angles, Vol. 4, No. 1 (March), pp. 5-6.
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Related Links
Building a Road (Intro to perpendicular from a point to a line)
2D Generalizations of Viviani's Theorem
Further generalizations of Viviani's Theorem (Using equi-inclined lines)
Parallelogram Distances (Same as 'Link to Parallelogram Distances' sketch above, but includes an Area variation)
Some 3D Generalizations of Viviani's Theorem
Clough's Theorem (a variation of Viviani) and some Generalizations
The Fermat-Torricelli Point (Rethinking Proof activity)
Airport Problem (Rethinking Proof activity)
Water Supply: Four Towns (Rethinking Proof activity: intro perpendicular bisectors)
Triangle Altitudes (Rethinking Proof activity)
Light Ray in a Triangle (Rethinking Proof activity)
Fermat-Torricelli Point Generalization
Carnot's (or Bottema's) Perpendicularity Theorem & Some Generalizations
A variation of Miquel's theorem and its generalization (uses equi-inclined lines)
A generalization of Neuberg's Theorem & the Simson line (using equi-inclined lines)
Some External Links
Viviani's theorem (Wikipedia)
Vincenzo Viviani (Wikipedia)
Vincenzo Viviani (MacTutor)
Viviani's Theorem (Cut The Knot)
Clough's theorem: the simplest proof (Cut The Knot)
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, 14 June 2025; updated 21 June 2025; 20 Nov 2025; 4 Dec 2025.