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 2*n*-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.

*Back to "Dynamic Geometry Sketches"*

*Back to "Student Explorations"*

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