Explore
Given below is a regular hexagon ABCDEF with a point P on diagonal BF. The areas of triangles FPA, BPC and DPE 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 three areas? Can you guess a relationship and make a conjecture?
3) Click on the 'Show Relationship between Areas' button to view the relationship between the three areas.
4) Check whether the relationship always holds by dragging P - also drag P to the outside of the regular hexagon. What do you notice? Does the result still hold?
Challenge
5) Can you explain why (prove that) the result observed above is true? Can you do it in more than one way?
(If stuck, see below for some Hints to possible ways of proving the result or compare your solution with some solutions).
Vreken's sum of triangular areas in a regular hexagon
Hints
6) Click on the 'Show Hints' button for some additional measurements that should give you some useful clues to a neat, elegant way of proving the result.
7) Alternatively, extend the alternate sides of the regular hexagon to form equilateral triangles & apply Viviani's theorem.
8) Were you able to prove the result in a different way to the solutions in 6) & 7)?
9) Compare your own logical explanations and proofs with these selected Solutions posted in the 'Romantics of Geometry' group on Facebook in August 2026.
Acknowledgement
This lovely hexagon area result is due to David Vreken from Dominion Christian School, Marietta GA, USA. He posted it as a problem in the 'Romantics of Geometry' group on Facebook in August 2026.
Comment
As illustrated in the dynamic sketch above, the result remains true even if P is dragged outside the regular hexagon and can be easily explained (proved) using the idea of 'directed areas' (see Coxeter & Greitzer, 1967).
References
Coxeter, H. S. M. & Greitzer, S. L. (1967). Directed Areas Excerpt from Geometry Revisited. MAA, Washington, pp. 51-54.
De Villiers, M. (2020). The Value of using Signed Quantities in Geometry. Learning & Teaching Mathematics, No. 29, pp. 30-34.
Related Links
Parallelogram Distances & Areas (Click 'Link to Areas')
Distances in an Equilateral Triangle (Viviani's theorem) (Rethinking Proof activity)
Sums of alternate triangular areas in regular & semi-regular side-2n-gons (Application of Viviani)
2D Generalizations of Viviani's Theorem to Polygons
Varignon Parallelogram Area (Rethinking Proof activity)
Area ratios of some polygons inscribed in quadrilaterals and triangles (Generalization of Varignon's theorem)
Parallel Lines (Thomsen's Hexagon) (Rethinking Proof activity; incl. of generalization)
Area Parallelogram Partition Theorem
International Mathematical Talent Search (IMTS) Area Ratio Problem Generalized
A Geometric Paradox Explained (Another variation of an IMTS problem)
Another parallelogram area ratio
Feynman Parallelogram Generalization
Feynman's Triangle: Some Generalizations & Variations
Feynman Triangle & Parallelogram Variations
Hippocrates' Theorem (470 – c. 410 BC)
Some Parallelo-hexagon Area Ratios
Easy Hexagon Explorations
Area Formula for Quadrilateral in terms of its Diagonals
The Area of an Orthodiagonal quadrilateral
Cross's theorem & generalizations to quadrilaterals
Sylvie's Theorem
Some Parallelo-hexagon Area Ratios
Bicentric Polygon Area Formula in terms of angles, r & R
Crossed Quadrilateral Properties
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula
A Surprising Constant Area Sum Involving Translating Figures
An Area Preserving Transformation: Shearing
Miscellaneous Dynamic Transformations (of Geometric Figures & Graphs)
External Links
Viviani's theorem (Wikipedia)
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, 24 August 2026; updated 27 August 2026.