MathDB

Let

d \geq 3

and let

A_1 \dots A_{d + 1}

be a simplex in

\mathbb{R}^d

. (A simplex is the convex hull of

d + 1

points not lying in a common hyperplane.) For every

i = 1, \dots , d + 1

let

O_i

be the circumcentre of the face

A_1 \dots A_{i - 1}A_{i+1}\dots A_{d+1}

, i.e.

O_i

lies in the hyperplane

A_1 \dots A_{i - 1}A_{i+1}\dots A_{d+1}

and it has the same distance from all points

A_1, \dots , A_{i-1}, A_{i+1}, \dots , A_{d+1}

. For each

i

draw a line through

A_i

perpendicular to the hyperplane

O_1 \dots O_{i-1}O_{i+1} \dots O_{d+1}

. Prove that either these lines are parallel or they have a common point.

Problem Statement