MathDB

Given a tetrahedron

A_1A_2A_3A_4

(not necessarily regulart). We shall call a point

N

in space Serve point, if it's six projection points on the six edges of the tetrahedron lie on one plane. This plane we denote it by

a (N)

and call the Serve plane of the point

N

. By

B_{ij}

denote, respectively, the midpoint of the edges

A_1A_j

1\le i <j \le 4

. For each point

M

, denote by

M_{ij}

the points symmetric to

M

with respect to

B_{ij},

1\le i <j \le 4

. Prove that if all points

M_{ij}

are Serve points, then the point

M

belongs to all Serve planes

a (M_{ij})

1\le i <j \le 4

Problem Statement