In analytical geometry, is the centroid of a simplex the mean of its vertices' coordinates?

Since there is a infinity of cases, proof by induction will be used.

A point's centroid is the point, regardless of the dimensionality of the space in which it is embedded.

A line segment's centroid is the arithmetic mean of its ends (vertices) coordinates , by simple examination, regardless of the dimensionality of the space in which it is embedded.

Those statements are the base of the induction, that is, they are the base proved specific cases.

Implicit in the definition of simplices (the singular is simplex) is that a simplex of $n+1$ dimensionality have one more vertex than a simplex of \(n) dimensionality.

A \(n\) dimensional simplex embedded in $m\geq n+1$ dimensional space, has $n+1$ vertices of dimensionality $m$ . If a point of the same dimensionality ( $m$ ) as the extant vertices were added to that simplex and which is different from those extant vertices, it would have the correct of vertices for a $n+1$ dimensional simplex with a dimensionality sufficient for its embedment.

It is sufficient to prove the statement for the first dimension as the proof for any remaining dimensions would be the same.

Using $c$ for centroid and $a$ for the new point:

$c_{n,1} =\frac{\sum_{i=1}^{n+1} v_{i,1}}{n+1} \implies (n+1)c_{n,1}=\sum_{i=1}^{n+1} v_{i,1}\implies a_1+\sum_{i=1}^{n+1} v_{i,1}\implies c_{n+1,1} =\frac{\sum_{i=1}^{n+2} v_{i,1}}{n+2}\ \therefore$