Can we assume that any interior point $z$ in a convex set $S\subseteq R^n $ be represented by $2$ points $x \in S$ and $y \in S$ such that $z = \lambda x +(1-\lambda)y $, where $x\neq y \neq z$ , and $\lambda \in (0,1)$?

What I understand is that every point $\in S$ can be represented as a convex combination of 2 points in $S$, and that any exterior point (say $v$) can't be constructed by 2 different points $x$ and $y$ ($x \neq y \neq v$) such that $v = \lambda x + (1-\lambda)y$ where $\lambda \in (0,1)$.

Intuitively, I think for any interior point it should be true unless the set only has 1 element.

How should I go about proving this seemingly trivial-looking result (if it is correct) or can I simply state that?