Exercise 13, § 3

Let $E$ be a normed space over the field $R$. Now prove that if $x, y, z$ are three linearly dependent points in $E$, one has

(Considering, if necessary, the points $x' = x, y' = y, z' = -x - y - z$, prove that one can reduce to the case where $x + \beta y + \gamma z = 0$ with $0 \leq \beta \leq 1, 0 \leq \gamma \leq 1$; then write $x + y = (1 - \beta)y - \gamma z, -(x + z) = \beta y - (1 - \gamma)z.$)