Consider a solid ADECBF consisting of two identical pyramids with the square ABCD as common base with centre I. A perspective representation of this solid is given in the appendix (to be returned with the answer sheet). All edges have length 1.\\
The space is referred to the orthonormal coordinate system ( $\mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AK } }$ ).

\begin{enumerate}
  \item a) Show that $\mathrm { IE } = \frac { \sqrt { 2 } } { 2 }$. Deduce the coordinates of points I, E and F.\\
b) Show that the vector $\vec { n } \left( \begin{array} { c } 0 \\ - 2 \\ \sqrt { 2 } \end{array} \right)$ is normal to the plane (ABE).\\
c) Determine a Cartesian equation of the plane (ABE).
  \item Let M be the midpoint of segment [DF] and N the midpoint of segment [AB].\\
a) Prove that the planes $( \mathrm { FDC } )$ and $( \mathrm { ABE } )$ are parallel.\\
b) Determine the intersection of planes (EMN) and (FDC).\\
c) Construct on the appendix (to be returned with the answer sheet) the cross-section of solid ADECBF by plane (EMN).
\end{enumerate}