Sums of alternate triangular areas in regular & semi-regular side-2n-gons

Exploration 1: Square
Given below is a square ABCD with an arbitrary point P inside. The areas of triangles APB, BPC, CPD and DPA are measured and displayed.
1) Drag P to see how the areas change. (Alternatively, click on the 'Animate Point P' button - to stop the animation, click again).
2) What relationship is there between these four areas? Can you guess a relationship and make a conjecture?
3) Click on the 'Show Sums of Alternate Areas' button to view a relationship between the four areas.
4) Check whether the relationship always holds by further dragging P. Also drag P to the outside of the square. What do you notice? Does the result still hold?

Challenge 1
5) Can you explain why (prove that) the result observed above is true for a square? Can you do it in more than one way?
(If stuck, click on the 'Show Proof Hint' buttons for some Hints to a possible way of proving the result).
6) Can you generalize further to other polygons?

Sums of alternate triangular areas in regular & semi-regular side-2n-gons

Exploration 2: Regular Hexagon
7) Click on the 'Link to regular hexagon' button to navigate to a new dynamic sketch with an arbitrary point P, and triangular areas, subtended from P to the sides, measured.
8) Click on the 'Show Sums of Alternate Areas' button to view a relationship between the six measured areas.
9) Check your observation in 8) by dragging P (or click on the 'Animate Point P' button).
10) Does your observation hold if P is dragged outside the hexagon? Formulate your observations as a conjecture.
Challenge 2
11) Can you explain why (prove that) the result you observed in 8) is true for a regular hexagon? Can you do it in more than one way?
(If stuck, click on the 'Show Proof Hint' button for some Hints to a possible way of proving the result or see Samson, 2012: pp. 30-32).
12) Can you generalize the result further to other polygons?

Directed or signed areas
In order to explain why (and prove that) the two results above remain true when P is outside the square or regular hexagon, requires use of 'directed' or 'signed' areas. In this case, it simply means that the area of a triangle changes sign when its orientation changes from anti-clockwise to clockwise (or vice versa). Read more about it in Coxeter & Greitzer (1967) in the references below.

Further Generalization
Interestingly these results not only generalize to any regular 2n-gon (e.g see Bogomolny, 1996-2018), but also to any semi-regular side-2n-gon which can be defined as any 2n-gon with an axis of symmetry through each pair of opposite angles (vertices), or equivalently, as any equilateral, tangential 2n-gon (see De Villiers, 2011). A rhombus is the most elementary example as shown below.

Exploration 3: Rhombus
13) Click on the 'Link to rhombus' button to navigate to a new dynamic sketch and click on the 'Show Sums of Alternate Areas' button to view a relationship between the measured areas.
14) Again check your observation in 13) by dragging P or clicking on the 'Animate Point P' button.
15) Do your observations still hold if the shape of the rhombus is changed or P is dragged outside the rhombus? Formulate your observations as a conjecture.
Challenge 3
16) Can you explain why (prove that) the result you observed in 13) is true for a rhombus? Can you do it in more than one way?
(Reflect on your proof for a square, and check if it applies to a rhombus. If still stuck, click on the 'Show Proof Hint' button for some Hints to a possible way of proving the result).
17) Can you generalize the result further to other semi-regular side-2n-gons?

Exploration 4: Semi-regular Side-hexagon
18) Click on the 'Link to semi-regular side-hexagon' button to navigate to a new dynamic sketch and click on the 'Show Sums of Alternate Areas' button to view a relationship between the measured areas.
19) Again check your observation in 18) by dragging P or clicking on the 'Animate Point P' button.
20) Do your observations still hold if the shape of the semi-regular side-hexagon is changed or P is dragged outside the semi-regular side-hexagon? Formulate your observations as a conjecture.
Challenge 4
21) Can you explain why (prove that) the result you observed in 18) is true for a semi-regular side-hexagon? Can you do it in more than one way?
(Reflect on your proof for a regular hexagon, and check if it applies to a semi-regular side-hexagon. If still stuck, click on the 'Show Proof Hint' button for some Hints to a possible way of proving the result).
22) Can you generalize the result further to other semi-regular side-2n-gons?

Exploration 5: Semi-regular Side-octagon
23) Click on the 'Link to convex s-r side-octagon' button to navigate to a new dynamic sketch showing a convex semi-regular side-octagon. Then click on the 'Show Sums of Alternate Areas' button to view a relationship between the measured areas.
24) Again check your observation in 23) by dragging P or clicking on the 'Animate Point P' button.
25) Do your observations still hold if the shape of the semi-regular side-octagon is changed or P is dragged outside the semi-regular side-octagon? Formulate your observations as a conjecture.
26) Click on the 'Link to crossed s-r side-octagon' button to navigate to a new dynamic sketch showing a crossed semi-regular side-octagon. Then click on the 'Show Sums of Alternate Areas' button to view a relationship between the measured areas.
27) In respect of your observation in 26), repeat steps 24) and 25).
Challenge 5
21) Can you explain why (prove that) the results you observed in 23) and 25) is true for both a convex and crossed semi-regular side-octagon? Can you do it in more than one way?
(Reflect on your proof for a rhombus, and check if it applies to a semi-regular side-octagon. If still stuck, click on the 'Show Proof Hint' button for some Hints to a possible way of proving the result).
22) Can you generalize the result further to other semi-regular side-2n-gons?

Comment
As is the case for regular 2n-gons, to provide a general proof for the semi-regular side-2n-gons for the results illustrated above, we need to distinguish between two different cases - for n even and n odd:
a) for n = 2 (rhombus), n = 4 (semi-regular side-octagon), n = 6 (semi-regular side-dodecagon), etc. the result follows easily from the constant distance (or width) between each pair of opposite, equal and parallel, sides
b) for n = 3 (semi-regular side-hexagon), n = 5 (semi-regular side-decagon), n = 7 (semi-regular side-14-gon), etc. the result follows from extending the alternate sides to form two congruent equi-angular n-gons, to which we can then simply apply a generalization of Viviani's theorem to equi-angular and/or equilateral polygons (see De Villiers, 2005; Abboud, 2010; Zhou, 2012; Alhajjar & Nasta, 2022).
(Footnote: The two equi-angular n-gons created by the extension of the alternate sides of a semi-regular side-2n-gon when n is odd can easily be shown to be equilateral - in other words, they are regular n-gons. This follows from the reflective symmetry around any of its main diagonals and/or the rotational symmetry of order n of the corresponding semi-regular side-2n-gon.)

Accepted Paper
A paper entitled 'Sums of alternate triangular areas in regular & semi-regular 2n-gons' regarding the results illustrated above has been accepted for publication in Mathematics in School (to appear on January or March 2027). All rights reserved.

References
Abboud, E. (2010). Viviani's Theorem and Its Extension. The College Mathematics Journal, Vol. 41, No. 3 (May), pp. 203-211.
Alhajjar, E. & Nasta, M. (2022). Viviani’s Theorem, Minkowski’s Theorem and Equiangular Polygons. BSU Undergraduate Mathematics Exchange, Vol. 16, No. 1 (Fall), pp. 66 – 72.
Bogomolny, A. (Dated 1996-2018). Equal Areas in Regular 2n-gons. Cut The Knot website.
Coxeter, H. S. M. & Greitzer, S. L. (1967). Directed Areas Excerpt from Geometry Revisited. MAA, Washington, pp. 51-54.
De Villiers, M. (2005). Crocodiles and Polygons (Generalization of Viviani to equi-angular polygons). Mathematics in School, pp. 2-4.
De Villiers, M. (2011). Equi-angled cyclic and equilateral circumscribed polygons. The Mathematical Gazette, 95(532), pp. 102-106.
De Villiers, M. (2020). The Value of using Signed Quantities in Geometry. Learning & Teaching Mathematics, No. 29, pp. 30-34.
Samson, D. (2012). Viviani’s Theorem – A Geometrical Diversion. Learning & Teaching Mathematics, No. 13, pp. 28-32.
Zhou, L. (2012). Viviani Polytopes and Fermat Points. College Mathematics Journal, 43 (4), pp. 309–312.

External Links
Viviani's theorem (Wikipedia)
Equal Areas in Regular 2n-gons (Cut The Knot)
Total Areas of Alternating Subtriangles in a regular 2n-gon (Mathematics Stack Exchange)
Regular hexagon divided into triangles (Mathematics Stack Exchange)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad & SA Mathematics Challenge Questions and worked solutions for past South African Mathematics Olympiad & Mathematics Challenge 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, 30 August 2026; updated 3 Sept 2026; 8 Sept 2026.