3D Generalisations of Viviani's theorem

Abstract

“The ambitious learner should carefully study a new fact; he should turn it over and over, consider it under various aspects, scrutinize it from all sides … Moreover, he should try to expand and enlarge any newly acquired knowledge by application, generalization, specialization, analogy, and in all other ways.” [1]Vincenzo Viviani, a 17th century mathematician, proved that the sum of the (perpendicular) distances from a point within an equilateral triangle to its sides is constant as shown in Figure 1. The theorem, named after him, generalises to polygons that are equilateral or equi-angled, or to 2n-gons with opposite sides parallel. Viviani's theorem is easily proved by summing the areas of triangles APB, BPC and CPA, equating to the area of the triangle ABC, and then simplifying to obtain h1 + h2 + h3 = H, where H is the altitude of ABC.

Removing a vertex from a tetrahedron

“…We shall now represent our result on the cube graphically by using the familiar theorem of the 17th century mathematician Vincenzo Viviani, namely that the sum of the distances from a point inside an equilateral triangle to the three sides of the triangle is constant, and equal to the height of the triangle (see [2] for more information). Let be the equilateral triangle which has vertices at , and .…”