\begin{exercice} \label{TSG3_E_eq_plan_trois_points_cube}
On considère un cube $ABCDEFGH$ d'arête 1 et le point $I$ tel que $3\vv{DI} = 2\vv{DC}$.
\begin{center}
 	\def \dx {-0.4}
 	\def \r {0.3}
	\begin{tikzpicture}[general, scale=2]
	%points
	\coordinate (A) at (0,0); \draw (A) node[below right] {$A$};
	\coordinate (B) at (\dx,-0.5); \draw (B) node[below left] {$B$};
	\coordinate (C) at (1+\dx,-0.5); \draw (C) node[below] {$C$};
	\coordinate (D) at (1,0); \draw (D) node[right] {$D$};
	\coordinate (E) at (0,1); \draw (E) node[above left] {$E$};
	\coordinate (F) at (\dx,0.5); \draw (F) node[left] {$F$};
	\coordinate (G) at (1+\dx,0.5); \draw (G) node[right] {$G$};
	\coordinate (H) at (1,1); \draw (H) node[above right] {$H$};
	\coordinate (I) at (1+2/3*\dx,-2/3*0.5); \draw (I) node[right] {$I$};
	%arêtes
	\draw (F)--(B)--(C)--(D)--(H)--(G)--(C); \draw (G)--(F)--(E)--(H);
	\draw[dashed] (A)--(B); \draw[dashed] (A)--(D); \draw[dashed] (A)--(E);
	%plan
	\fill[color=A1, opacity=0.3] (F)--(H)--(I);
	\end{tikzpicture}
\end{center}
En se plaçant dans le repère orthonormé $(A\,; \vv{AB}, \vv{AD}, \vv{AE})$, déterminer une équation cartésienne du plan $(FHI)$.
\end{exercice}


