MathDB

Let

n

be a positive integer. An n-\emph{simplex} in

\mathbb{R}^n

is given by

n+1

points

P_0, P_1,\cdots , P_n

, called its vertices, which do not all belong to the same hyperplane. For every

n

-simplex

\mathcal{S}

we denote by

v(\mathcal{S})

the volume of

\mathcal{S}

, and we write

C(\mathcal{S})

for the center of the unique sphere containing all the vertices of

\mathcal{S}

. Suppose that

P

is a point inside an

n

-simplex

\mathcal{S}

. Let

\mathcal{S}_i

be the

n

-simplex obtained from

\mathcal{S}

by replacing its

i^{\text{th}}

vertex by

P

. Prove that :

\sum_{j=0}^{n}v(\mathcal{S}_j)C(\mathcal{S}_j)=v(\mathcal{S})C(\mathcal{S})

Problem Statement