MathDB

It is given a tetrahedron with vertices

A,B,C,D

.
(a) Prove that there exists a vertex of the tetrahedron with the following property: the three edges of that tetrahedron through that vertex can form a triangle. (b) On the edges

DA,DB

and

DC

there are given the points

M,N

and

P

for which:

DM=\frac{DA}n,\enspace DN=\frac{DB}{n+1}\enspace DP=\frac{DC}{n+2}

where

n

is a natural number. The plane defined by the points

M,N

and

P

\alpha_n

. Prove that all planes

\alpha_n

(n=1,2,3,\ldots)

pass through a single straight line.

Problem Statement