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
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. Drag point *P* or any of the red
vertices to explore the results.

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.

Equilateral pentagon distances

Equi-angled pentagon distances

Click on the **Show Hint** button 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.

Parallel-hexagon distances

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**

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**

An interesting variation of Viviani's theorem was experimentally discovered by a schoolboy, Clough, in 2003 and is available at: *Clough's Theorem and some generalizations*.

This page uses

*Back to "Dynamic Geometry Sketches"*

Updated by Michael de Villiers, 5 May 2013.