We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$. a) Prove that, if we fix $z\in\mathbf{C}$ and $(a,b,c)\in\mathbf{C}^3$: $$\max\left\{|z'-z| / z'\in\widehat{abc}\right\} = \max(|z-a|,|z-b|,|z-c|)$$ b) Deduce from this a simple expression for $\delta(\widehat{abc})$.
We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$.
a) Prove that, if we fix $z\in\mathbf{C}$ and $(a,b,c)\in\mathbf{C}^3$:
$$\max\left\{|z'-z| / z'\in\widehat{abc}\right\} = \max(|z-a|,|z-b|,|z-c|)$$
b) Deduce from this a simple expression for $\delta(\widehat{abc})$.