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\partie{Homework \#6}

Let $ABC$ be an equilateral triangle with side 3 cm. For the figure, $A$ will be placed on top of the triangle and on
top of the paper sheet, with $B$ on the right-hand side and $C$ on the left-hand side.
\begin{enumerate}
\item Place the point $D$ such that $\vect{AD}=\vect{AB}+\vect{AC}$.
\item Place the point $E$ such that $\vect{AE}=3\vect{AC}$.
\item Place the point $F$ such thate $\vect{AF}=2\vect{AB}+\vect{AC}$.
\item Place the point $M$ such that $\vect{MA}=-2\vect{AB}$.
\item Let $N$ be the point such that $\vect{NA}=3\vect{BA}$.
    \begin{enumerate}
    \item Decompose $\vect{AN}$ using only $\vect{AB}$ and $\vect{AC}$
    \item Place the point $N$.
    \end{enumerate}
\item Let $P$ be the point such that $\vect{PA}+\vect{PB}=4\vect{CA}+\vect{BA}$.
    \begin{enumerate}
    \item Decompose $\vect{AP}$ using only $\vect{AB}$ and $\vect{AC}$
    \item Place the point $P$.
    \end{enumerate}
\item Let $Q$ be the point such that $\vect{QA}+\vect{QB}+\vect{QC}=8\vect{CA}+2\vect{BA}$.
    \begin{enumerate}
    \item Decompose $\vect{AQ}$ using only de $\vect{AB}$ and $\vect{AC}$
    \item Place the point $Q$.
    \end{enumerate}
\item Using the previous questions and Chasles' relation, decompose $\vect{DB}$, $\vect{DP}$ and
$\vect{DQ}$ using
$\vect{AB}$ and $\vect{AC}$. What can you deduce about the points $B$, $D$, $P$,$Q$ ?
\item \begin{enumerate}
    \item Place the point $I$, midpoint of the segment $[FP]$.
    \item Find out a relation between the vectors $\vect{FI}$ and $\vect{FP}$.
    \item Write $\vect{FP}$ and $\vect{FI}$ using $\vect{AB}$ and $\vect{AC}$.
    \item Use $\vect{AF}$ and $\vect{FI}$ to decompose $\vect{AI}$ with $\vect{AB}$ and $\vect{AC}$.
    \item Prove that $\vect{AD}=\frac{2}{3}\vect{AI}$. What can you deduce about the point $D$ in the triangle $AEN$ ?
    \item Deduce from the previous questions the intersection point of $(ND)$ and $(AE)$ and the intersection point of
the lines $(ED)$ and $(AN)$.
    \end{enumerate}
\end{enumerate}

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