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 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 $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. a) Let $a \in \mathbf{C}$ and $\theta \in \mathbf{R}$. Prove that the image $z'$ of the complex number $z$ by the reflection whose axis is the line passing through $a$ and directed by $\mathrm{e}^{\mathrm{i}\theta}$ satisfies the relation: $$z' - a = \mathrm{e}^{2\mathrm{i}\theta} \overline{(z-a)}$$ b) Establish a relation analogous to that of the previous question between a complex number $z$ and its image $z'$ by the homothety with center $a$ and ratio $\rho > 0$. c) Prove that $\phi_0$ is the composition of a reflection whose axis we will specify and a homothety with strictly positive ratio to be specified and whose center belongs to the axis of the reflection. Prove an analogous property for $\phi_1$. Are these decompositions unique?
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 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 $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$.
a) Let $a \in \mathbf{C}$ and $\theta \in \mathbf{R}$. Prove that the image $z'$ of the complex number $z$ by the reflection whose axis is the line passing through $a$ and directed by $\mathrm{e}^{\mathrm{i}\theta}$ satisfies the relation:
$$z' - a = \mathrm{e}^{2\mathrm{i}\theta} \overline{(z-a)}$$
b) Establish a relation analogous to that of the previous question between a complex number $z$ and its image $z'$ by the homothety with center $a$ and ratio $\rho > 0$.
c) Prove that $\phi_0$ is the composition of a reflection whose axis we will specify and a homothety with strictly positive ratio to be specified and whose center belongs to the axis of the reflection. Prove an analogous property for $\phi_1$. Are these decompositions unique?