In space, we consider a pyramid SABCE with square base ABCE with centre O. Let D be the point in space such that ( O ; $\overrightarrow { \mathrm { OA } } , \overrightarrow { \mathrm { OB } } , \overrightarrow { \mathrm { OD } }$ ) is an orthonormal frame. The point S has coordinates $( 0 ; 0 ; 3 )$ in this frame.

\section*{Part A}
\begin{enumerate}
  \item Let U be the point on the line $( \mathrm { SB } )$ with height 1. Construct the point U on the attached figure in appendix 1.
  \item Let V be the intersection point of the plane (AEU) and the line (SC). Show that the lines (UV) and (BC) are parallel. Construct the point V on the attached figure in appendix 1.
  \item Let K be the point with coordinates $\left( \frac { 5 } { 6 } ; - \frac { 1 } { 6 } ; 0 \right)$. Show that K is the foot of the altitude from U in the trapezoid AUVE.
\end{enumerate}

\section*{Part B}
In this part, we admit that the area of the quadrilateral AUVE is $\frac { 5 \sqrt { 43 } } { 18 }$.
\begin{enumerate}
  \item We admit that the point $U$ has coordinates $\left( 0 ; \frac { 2 } { 3 } ; 1 \right)$. Verify that the plane (EAU) has equation $3 x - 3 y + 5 z - 3 = 0$.
  \item Give a parametric representation of the line (d) perpendicular to the plane (EAU) passing through the point $S$.
  \item Determine the coordinates of H, the intersection point of the line (d) and the plane (EAU).
  \item The plane (EAU) divides the pyramid (SABCE) into two solids. Do these two solids have the same volume?
\end{enumerate}