Richard P. answered 08/01/18
Fairfax County Tutor for HS Math and Science
A neat way of working this problem is to note that a tetrahedron can be inscribed into a cube. Drawings of this construction can be found in some advanced textbooks. Analytically,if the eight corners of a cube are
(c,c,c), (-c,c,c),(-c-c,c),(c, -c,c),(c,c,-c),(-c,c,-c),(-c,-c,-c),(c,-c,-c) then the corners of the inscribed tetrahedron can be
(c,c,c), (-c,-c,c),(-c,c,-c), (c,-c,-c) . With the choice of (c,c,c) as the top point , the vertices of the equilateral triangle base
are: (-c,c, -c), (-c,-c,c) , (c, -c, -c). The point at the center of this triangle is (-c/3 , -c/3, -c/3) . The distance formula can be use to get the distance between this center point and the tip corner = 4 c sqrt(3)/3. The edge length of the tetrahedron , s, is s = 2 c sqrt(2). so c = s / (2 sqrt(2)) . Substituting for c gives the desired distance as
s sqrt(6) /3
Is it necessary that every tetrahedron has equilateral triangular base??
The word tetrahedron refers to one of the 5 Platonic solids. The faces of these solids are equilateral. So the four faces of the tetrahedron are equilateral triangles.