Viviani's theorem

The set of Viviani, named after the Italian mathematician Vincenzo Viviani (1622-1703), is a simple statement of equilateral triangles:

Is an arbitrary point in the interior of an equilateral triangle, the sum of the distances from the sides of this point is constant:

It is the height of the triangle and the Inkreisradius.

This can be made clear geometrically simple. The surface of the equilateral triangle is equal to the sum of the areas of the color-coded triangles.

Applies to the surface of the equilateral triangle ABC, wherein the base side and the amount should be.

The sum of the areas of the color-coded triangles.

So the following applies:

Thus the assertion follows.

The set of Viviani can be generalized to equilateral and equiangular polygons even on.