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 2n-gons
with opposite sides parallel.

Investigate: Explore the generalization of Viviani's theorem to 3D by making use of the analogies between 2D and 3D; that is consider what the equivalent analogies in 3D are for 2D concepts like triangle, side and area. Specifically, it might help to carefully consider the 2D proof and its explanation of the result in 2D, and then to consider the analogous argument in 3D. Also consider generalizing the various 2D generalizations of Viviani's theorem to 3D.