  In this paper we present short algebraic proofs of the Linear
Conway--Gordon--Sachs and the Linear van Kampen--Flores theorems in the spirit
of the Radon theorem on convex hulls.
  {\bf Theorem.} {\it Take any $n+3$ general position points in $\mathbb{R}^n$.
If $n$ is odd, then there are two linked $(n+1)/2$-simplices with the vertices
at these points. If $n$ is even, then one can choose two disjoint
$(n+2)/2$-tuples such that the interiors $(n/2)$-simplices with the vertices at
these $(n+2)/2$-tuples intersect each other.}
  This theorem is interesting even in case of small dimensions.
