Generalizing Viviani's theorem to 3D

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.

Only if stuck, or to check go to 3D Generalizations of Viviani's theorem.