4-dimensional simplex

[fleksy]

In a 4-dimensional Euclidean space, there is a simplex, with given lengths of all the edges aij = distance(Ai,Aj). Find a distance between gravity centers of sides, opposite to each other. Notice: here, we call "a side" any kind of non-internal element, like a vertex (a 0-dimensional side), an edge (1-dim side), a triangular face (a 2-dim side), or a tetrahedron (3-dim side). Sides are "opposite", if a sum of their dimensions is equal to dimensionality of a space minus one (4-1=3 in this case). A vertex is opposite to a some a tetrahedron, a triangle - to some a segment. A gravity center of a segment is its center, that of a triangle - a point of intersection of its medians.

 

[stapel]

In a 4-dimensional Euclidean space, there is a simplex, with given lengths of all the edges aij = distance(Ai,Aj). Find a distance between gravity centers of sides, opposite to each other.

Notice: here, we call "a side" any kind of non-internal element, like a vertex (a 0-dimensional side), an edge (1-dim side), a triangular face (a 2-dim side), or a tetrahedron (3-dim side). Sides are "opposite", if a sum of their dimensions is equal to dimensionality of a space minus one (4-1=3 in this case).

A vertex is opposite to a some a tetrahedron, a triangle - to some a segment.

A gravity center of a segment is its center, that of a triangle - a point of intersection of its medians.

What are your thoughts? What have you tried? How far have you gotten? Where are you stuck?

Please be complete. Thank you!