The plane is equipped with an orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). The points $\mathrm{A}$, $\mathrm{B}$ and C have affixes respectively $a = -4$, $b = 2$ and $c = 4$.
We consider the three points $\mathrm{A}^{\prime}$, $\mathrm{B}^{\prime}$ and $\mathrm{C}^{\prime}$ with affixes respectively $a^{\prime} = \mathrm{j}a$, $b^{\prime} = \mathrm{j}b$ and $c^{\prime} = \mathrm{j}c$ where j is the complex number $-\frac{1}{2} + \mathrm{i}\frac{\sqrt{3}}{2}$. a. Give the trigonometric form and the exponential form of j. Deduce the algebraic and exponential forms of $a^{\prime}$, $b^{\prime}$ and $c^{\prime}$. b. The points $\mathrm{A}$, $\mathrm{B}$ and C as well as the circles with center O and radii 2, 3 and 4 are represented on the graph provided in the Appendix. Place the points $\mathrm{A}^{\prime}$, $\mathrm{B}^{\prime}$ and $\mathrm{C}^{\prime}$ on this graph.
Show that the points $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ are collinear.
We denote M the midpoint of segment $[\mathrm{A}^{\prime}\mathrm{C}]$, N the midpoint of segment $[\mathrm{C}^{\prime}\mathrm{C}]$ and P the midpoint of segment $[\mathrm{C}^{\prime}\mathrm{A}]$. Prove that triangle MNP is isosceles.
The plane is equipped with an orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). The points $\mathrm{A}$, $\mathrm{B}$ and C have affixes respectively $a = -4$, $b = 2$ and $c = 4$.
\begin{enumerate}
\item We consider the three points $\mathrm{A}^{\prime}$, $\mathrm{B}^{\prime}$ and $\mathrm{C}^{\prime}$ with affixes respectively $a^{\prime} = \mathrm{j}a$, $b^{\prime} = \mathrm{j}b$ and $c^{\prime} = \mathrm{j}c$ where j is the complex number $-\frac{1}{2} + \mathrm{i}\frac{\sqrt{3}}{2}$.\\
a. Give the trigonometric form and the exponential form of j. Deduce the algebraic and exponential forms of $a^{\prime}$, $b^{\prime}$ and $c^{\prime}$.\\
b. The points $\mathrm{A}$, $\mathrm{B}$ and C as well as the circles with center O and radii 2, 3 and 4 are represented on the graph provided in the Appendix. Place the points $\mathrm{A}^{\prime}$, $\mathrm{B}^{\prime}$ and $\mathrm{C}^{\prime}$ on this graph.
\item Show that the points $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ are collinear.
\item We denote M the midpoint of segment $[\mathrm{A}^{\prime}\mathrm{C}]$, N the midpoint of segment $[\mathrm{C}^{\prime}\mathrm{C}]$ and P the midpoint of segment $[\mathrm{C}^{\prime}\mathrm{A}]$. Prove that triangle MNP is isosceles.
\end{enumerate}