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In this paper we present short algebraic proofs of the LinearConway--Gordon--Sachs and the Linear van Kampen--Flores theorems in the spiritof 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 verticesat 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 atthese $(n+2)/2$-tuples intersect each other.}  This theorem is interesting even in case of small dimensions.

Algebraic proofs of linear versions of the Conway--Gordon--Sachs theorem and the van Kampen--Flores theorem