The following little investigation is suitable for learners at approximately Grades 8-9. It is a 'Further Explore' activity from the Distances in an Equilateral Triangle (Viviani's theorem) investigation in my "Rethinking Proof with Sketchpad" book, which is now available for free as PDF to download at ResearchGate.
Suggestion: While this activity can be done independently, viewers & users are advised to first do the Distances in an Equilateral Triangle (Viviani's theorem) investigation mentioned above as this activity naturally connects to and follows on that activity.
Explore
1) ABCD is a parallelogram. Drag point P and observe the sum of the (perpendicular) distances from P to the sides. What do you notice?
2) Drag any of the red vertices of ABCD and then repeat step 1).
3) Drag P outside ABCD. What do you notice about the sum of the distances?
4) Formulate a conjecture on the basis of your observations.
Parallelogram Distances & Areas
Explanation
5) Can you logically explain why (prove that) your observations in 4) above are true?
(Note: You may not assume that the paths FPG and HPI are straight (even though they may appear to be so, you have to prove they are straight).
6) Read De Villiers (2020) below, or Google the concept of 'directed distances', and use it to ensure that your conjecture and proof also holds when P is dragged outside the parallelogram.
Further Generalization
7) Can you further generalize to hexagons, octogons, etc.?
8) In regard to 7), see the following related results, 2D Generalizations of Viviani's Theorem.
9) Can you generalize the parallelogram distances result to 3D?
(Check your answer to the above in De Villiers (2013, p. 443).
References
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
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 in Learning & Teaching Mathematics, No. 29, 2020, pp. 30-34.
Note: This paper describes how to use the above 'sum-of-distances' result for a parallelogram (and/or Viviani's theorem itself) to introduce students to the concept of 'directed distances' (and 'directed' areas).
Converse
9) Further investigate the converse of the parallelogram result, namely: If the sum of the (perpendicular) distances from an arbitrary point P to the sides of a quadrilateral is constant, then the quadrilateral is a parallelogram.
10) If true, can you prove it? If false, can you provide a counter-example?
11) Check your answer to 9) & 10) at any on these: Viviani's Theorem: Parallelogram Extension, Josefsson (2024, p. 79), or Shirali (2019, pp. 56-57).
References related to Converse
Josefsson, M. (2024). Characterization of Parallelograms, Part 2. International Journal of Geometry, Vol. 13 (2024), No. 3, pp. 65 - 93.
Shirali, S. (2019). How to prove it. At Right Angles, Issue 5 (Nov), pp. 51–57.
Area Variation
12) Click on 'Link to areas' button to navigate to a new sketch.
13) In the new sketch, click on 'Show Areas' button. What do yout notice? Check your observations by dragging & formulate a conjecture.
Challenge
14) Can you explain why (prove that) your observation in 13) above is true?
15) Can you extend the Area Variation result to the case when P is dragged outside the paralellogram by using the idea of 'directed' or 'signed distances' as explained in De Villiers (2020)?
16) Does the area result generalize further to higher polygons with opposite sides parallel? Explore & explain your observations.
Use in Mathematics Competitions
The area and distances results shown above (including Viviani's theorem), and variations of them, are often used in mathematics competitions. Here is just one example: Given a rectangle as shown below with dimensions 6 x 4, find the total area of the two shaded triangles.
Note: This particular problem has been used in the first rounds of South African Mathematics Challenge and South African (Junior) Mathematics Olympiad. It also appears from time to time in the Daily Math Puzzle at the top of the Dynamic Geometry Sketches title page.
Related Links
Distances in an Equilateral Triangle (Viviani's theorem) (Rethinking Proof activity)
Building a Road (Intro to perpendicular as shortest distance)
2D Generalizations of Viviani's Theorem
Some 3D Generalizations of Viviani's Theorem
Further generalizations of Viviani's Theorem (using equi-inclined lines)
Clough's Theorem (a variation of Viviani)
Area Parallelogram Partition Theorem
Modelling: Jannie's Father's Farm Problem
Another parallelogram area ratio
Golden Parallelogram
Similar Parallelograms: A Generalization of a Golden Rectangle property
A generalization of a Parallelogram Theorem to Parallelo-hexagons, Hexagons and 2n-gons in general
Euclid 1-43 Parallelogram Area Theorem
Feynman Parallelogram Generalization
International Mathematical Talent Search (IMTS) Problem Generalized
A Geometric Area Paradox Explained (Another variation of an IMTS problem)
Area ratios of some parallel-polygons inscribed in quadrilaterals and triangles
A Surprising Constant Area Sum Involving Translating Figures
Trio of Parallelograms
An associated result of the Van Aubel configuration and some generalizations
A Rectangle Angle Trisection Result
Crossed Quadrilateral Area
Exploring parallelogram properties
A generalization of Neuberg's Theorem and the Simson line
Equi-inclined Lines Problem
A variation of Miquel's theorem and its generalization
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula
External Links
Viviani's theorem (Wikipedia)
Viviani's Theorem (Cut The Knot)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Challenge (South African National Primary Olympiad)
SA Mathematics Olympiad (South African National Senior 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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Created by Michael de Villiers, 23 July 2020; updated 8 March 2021; 20 Oct 2024; 1 Dec 2025.