Definition 1: The convex hull of a subset $S$ of $\mathbb{R}^n$ is the smallest convex subset of $\mathbb{R}^n$ that contains $S$. I denote it by the symbol $\mbox{Conv}(S)$. If $S$ is a finite set, i.e. $S=\{g_1,\dots, g_k\}$ I will write $\mbox{Conv}(g_1,\dots, g_k)$.
Definition 2: Let $C$ be a convex subset of $\mathbb{R}^n$ and let $x\in C$. I say that $x$ is an extreme point of $C$ if $x$ cannot be expressed as a strict convex combination of two distinct points $a,b$ of $C$ with $a\ne x$ and $b\ne x$.
Proposition: if $g_1,\dots, g_k\in\mathbb{R}^n$ are points and $x$ is not an extreme point of $$C := \mbox{Conv}(g_1,\dots,g_k,x)$$ then $$\mbox{Conv} (g_1, \dots, g_k, x) = \mbox{Conv} (g_1, \dots, g_k)$$
I suspect that this proposition is true and I'm trying to prove it. The inclusion $\supseteq$ is trivial. I have to prove the other inclusion, i.e. $\subseteq$. It sufficies to prove that $x\in Conv(g_1,\dots, g_k)$. By the hypothesis we have that there exist two different points $a,b\in C$, with $a\ne x$ and $b\ne x$, and a scalar $0<\theta<1$ such that $x=\theta a+(1-\theta)b$. Since $a, b\in C$, we have that there exist
$$\mu_1,\dots, \mu_k,\mu_{k+1},\eta_1,\dots,\eta_k,\eta_{k+1}\in\mathbb{R}_0^+$$
such that $\sum_{i=1}^{k+1}\mu_i=1$, $\sum_{i=1}^{k+1}\eta_i=1$ and $a=\sum_{i=1}^{k+1}\mu_ig_i$, $b=\sum_{i=1}^{k+1}\eta_ig_i$. Then I have
$$x=\sum_{i=1}^k\theta\mu_i g_i+\sum_{i=1}^k(1-\theta)\eta_ig_i+\theta\mu_{k+1}x+(1-\theta)\eta_{k+1}x$$
i.e.,
$$(1-\theta\mu_{k+1}-(1-\theta)\eta_{k+1})x=\sum_{i=1}^k\theta\mu_i g_i+\sum_{i=1}^k(1-\theta)\eta_ig_i$$
Now if I show that $1-\theta\mu_{k+1}-(1-\theta)\eta_{k+1}\ne 0$ I would finish the proof. But I don't know how to prove it. Can anyone help me please?