Viviani's theorem

Viviani's theorem is the theorem named after Vincenzo Viviani stating that the sum of the distances from a point to the sides of an equilateral triangle equals the length of its altitude.

The theorem can be extended to equilateral polygons and equiangular polygons. :"The sum of distances from a point to the side lines of an equiangular polygon does not depend on the point and is that polygon's invariant."

Proof

This theorem can be easily proven by comparing areas of triangles. Let ABC be a equilateral triangle and P its interior point. Let "s" denote length of its side, "h" altitude and "l", "m", "n" distances of point P from sides. Then

:$S(ABC)\; =\; S(ABP)\; +\; S(ACP)\; +\; S(BCP),$

:$frac\{s\; h\}\{2\}\; =\; frac\{s\; l\}\{2\}\; +\; frac\{s\; m\}\{2\}\; +\; frac\{s\; n\}\{2\},$

:$h\; =\; l\; +\; m\; +\; n$

and that is what we wanted.

External links

* [http://www.cut-the-knot.org/Curriculum/Geometry/Viviani.shtml Viviani's Theorem: What is it?] at Cut the knot.
* [http://demonstrations.wolfram.com/VivianisTheorem/ Viviani's Theorem] by Jay Warendorff, The Wolfram Demonstrations Project.
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