Throughout the problem, the set $\mathbf{C}$ of complex numbers is considered as the Euclidean affine plane equipped with its canonical orthonormal coordinate system $(0,1,i)$ (where $i^2=-1$). 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 $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$. We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. What is the image of a filled triangle $\widehat{abc}$ by $\phi_0$ and by $\phi_1$? Determine $\phi_0(\tau)$ and $\phi_1(\tau)$.
Throughout the problem, the set $\mathbf{C}$ of complex numbers is considered as the Euclidean affine plane equipped with its canonical orthonormal coordinate system $(0,1,i)$ (where $i^2=-1$). 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 $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$. We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$.
What is the image of a filled triangle $\widehat{abc}$ by $\phi_0$ and by $\phi_1$? Determine $\phi_0(\tau)$ and $\phi_1(\tau)$.