Viviani's theorem is named after Vincenzo Viviani, a 17th century mathematician, who was a student of Evangelista Torricelli, the inventor of the barometer. The theorem states the surprising result that the sum of the (perpendicular) distances from a point to the sides of an equilateral triangle is constant. The theorem generalizes to (convex) polygons that are equilateral or equi-angled, or to 2n-gons with opposite sides parallel as shown respectively by the three interactive sketches below.
A free learning activity with a PDF worksheet that guides learners to discover and formulate Viviani's theorem, and to explain why (prove) that it is true, together with the further explorations below, is available at: Distances in an Equilateral Triangle (Viviani's theorem).
"The process of generalization, instead of leading from elements to classes, leads from classes to classes ... we shall regard abstraction as class formation, and generalization as class extension ...." - Zoltan Dienes (1961, pp. 282; 296). On Abstraction and Generalization. Harvard Educational Review, 31(3), pp. 281-301.
2D Generalizations of Viviani's Theorem to Polygons
Investigate
1) Drag point P or any of the red vertices to explore the results in the dynamic sketch above.
2) Use the LINK buttons in the sketch above to move to a) a hexagon with opposite sides parallel and b) an equi-angular pentagon.
3) Click on the Show Hint button in the equi-angular pentagon page for a hint to constructing a logical
explanation (proof) for the above result, or if you're really stuck, go to my 2005 paper in Mathematics in School at: Crocodiles
and Polygons.
4) Note that the results hold even when P is outside the polygon, or outside a pair of parallel lines, provided we regard distances respectively falling completely outside the polygon or outside the parallel lines as negative; in other words using directed distances (or equivalently, vectors). However, Sketchpad does not measure 'negative' distances, so dragging P outside will appear to no longer give a constant sum.
Further Generalizations with Equi-inclined lines
Viviani's theorem can be even further generalized by constructing lines to the sides of the above sets of polygons so that they form equal angles with the sides as shown with a dynamic sketch at Further generalizations of Viviani's theorem.
The theorem also generalizes to 3D as shown at 3D Generalizations of Viviani's Theorem.
A Variation on Viviani: Clough's Theorem
An interesting variation of Viviani's theorem was experimentally discovered by a Bishops Diocesan College schoolboy, Clough, in 2003 and is available at: Clough's Theorem and some generalizations.
Directed Distances
As mentioned in 4) above, Viviani's theorem and its various generalizations provide a good context for introducing talented students to directed distances when considering the case when the point P moves outside the polygon. For example, read my paper The Value of using Signed Quantities in Geometry in Learning & Teaching Mathematics, Dec 2020.
Reference
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Related Links
Distances in an Equilateral Triangle (Viviani's theorem) (Rethinking Proof activity)
Further generalizations of Viviani's Theorem (Using equi-inclined lines)
3D Generalizations of Viviani's Theorem
Clough's Theorem and some generalizations (A variation of Viviani)
Parallelogram Distances & Areas (intro to directed distances)
Vreken's sum of triangular areas in a regular hexagon (Application of Viviani)
Sums of alternate triangular areas in regular & semi-regular 2n-gons (Application of Viviani)
A variation of Miquel's theorem and its generalization
A generalization of Neuberg's Pedal Theorem & the Simson-Wallace line
A generalization of Neuberg's Pedal Theorem to polygons (Using equi-inclined lines)
Airport Problem (Rethinking Proof activity)
A Generalization of the British Flag theorem to Semi-Regular Angle-2n-gons
Sum of Squared Distances related to Incircles & Circumcircles
Sum of Squared Distances to Vertices of Polygons
Sum of powers of distances to vertices and sides
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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Free Download of Geometer's Sketchpad & Learning/Instructional Modules on various topics
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