2D Generalizations of Viviani's Theorem
Viviani

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 learning activity with a 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 given in my book Rethinking Proof with Sketchpad.

"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.

 

Investigate
1) Drag point P or any of the red vertices to explore the results.
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 equilateral pentagon one for 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.

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Created by Michael de Villiers, 2011; updated 5 May 2013; 14 April 2019; 8 March 2021.