Is it true that the circumcenter of an $n$-simplex, when in the simplex's interior, is a convex combination of the midpoints of the edges?

Question

Is the following true?:

Suppose we have an $n$-simplex such that the circumcenter of the simplex lies inside.

Is it true that the circumcenter is a convex combination of the midpoints of the edges of the simplex?

In 2D and 3D this seems to be true but I am not sure about higher dimensions. Maybe there is even a counterexample for 2D and 3D.