``In a non-equilateral triangle, the Euler line is the line that passes through the following three points:
the center of the circumscribed circle of this triangle (circle passing through the three vertices of this triangle).
the centroid of this triangle located at the intersection of the medians of this triangle.
the orthocenter of this triangle located at the intersection of the altitudes of this triangle''.
The purpose of the exercise is to study an example of an Euler line. We consider a cube ABCDEFGH with side length one unit. The space is equipped with the orthonormal coordinate system $( \mathrm { A } ; \overrightarrow { \mathrm { AB } } ; \overrightarrow { \mathrm { AD } } ; \overrightarrow { \mathrm { AE } } )$. We denote I the midpoint of segment [AB] and J the midpoint of segment [BG].
Give without justification the coordinates of points A, B, G, I and J.
a. Determine a parametric representation of the line (AJ). b. Show that a parametric representation of the line (IG) is: $$\left\{ \begin{aligned}
x & = \frac { 1 } { 2 } + \frac { 1 } { 2 } t \\
y & = t \\
z & = t
\end{aligned} \text { with } t \in \mathbb { R } . \right.$$ c. Prove that the lines (AJ) and (IG) intersect at a point S with coordinates $S \left( \frac { 2 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.
a. Show that the vector $\vec { n } ( 0 ; - 1 ; 1 )$ is normal to the plane (ABG). b. Deduce a Cartesian equation of the plane (ABG). c. We admit that a parametric representation of the line (d) with direction vector $\vec { n }$ and passing through the point K with coordinates $\left( \frac { 1 } { 2 } ; 0 ; 1 \right)$ is: $$\left\{ \begin{array} { l }
x = \frac { 1 } { 2 } \\
y = - t \quad \text { with } t \in \mathbb { R } . \\
z = 1 + t
\end{array} \right.$$ Show that this line (d) intersects the plane (ABG) at a point L with coordinates $L \left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$. d. Show that the point L is equidistant from the points $\mathrm { A } , \mathrm { B }$ and G.
Show that the triangle ABG is right-angled at B.
a. Identify the center of the circumscribed circle, the centroid and the orthocenter of triangle ABG (no justification is expected). b. Verify by calculation that these three points are indeed collinear.
``In a non-equilateral triangle, the Euler line is the line that passes through the following three points:
\begin{itemize}
\item the center of the circumscribed circle of this triangle (circle passing through the three vertices of this triangle).
\item the centroid of this triangle located at the intersection of the medians of this triangle.
\item the orthocenter of this triangle located at the intersection of the altitudes of this triangle''.
\end{itemize}
The purpose of the exercise is to study an example of an Euler line.
We consider a cube ABCDEFGH with side length one unit.
The space is equipped with the orthonormal coordinate system $( \mathrm { A } ; \overrightarrow { \mathrm { AB } } ; \overrightarrow { \mathrm { AD } } ; \overrightarrow { \mathrm { AE } } )$.
We denote I the midpoint of segment [AB] and J the midpoint of segment [BG].
\begin{enumerate}
\item Give without justification the coordinates of points A, B, G, I and J.
\item a. Determine a parametric representation of the line (AJ).\\
b. Show that a parametric representation of the line (IG) is:
$$\left\{ \begin{aligned}
x & = \frac { 1 } { 2 } + \frac { 1 } { 2 } t \\
y & = t \\
z & = t
\end{aligned} \text { with } t \in \mathbb { R } . \right.$$
c. Prove that the lines (AJ) and (IG) intersect at a point S with coordinates $S \left( \frac { 2 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.\\
\item a. Show that the vector $\vec { n } ( 0 ; - 1 ; 1 )$ is normal to the plane (ABG).\\
b. Deduce a Cartesian equation of the plane (ABG).\\
c. We admit that a parametric representation of the line (d) with direction vector $\vec { n }$ and passing through the point K with coordinates $\left( \frac { 1 } { 2 } ; 0 ; 1 \right)$ is:
$$\left\{ \begin{array} { l }
x = \frac { 1 } { 2 } \\
y = - t \quad \text { with } t \in \mathbb { R } . \\
z = 1 + t
\end{array} \right.$$
Show that this line (d) intersects the plane (ABG) at a point L with coordinates $L \left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$.\\
d. Show that the point L is equidistant from the points $\mathrm { A } , \mathrm { B }$ and G.\\
\item Show that the triangle ABG is right-angled at B.\\
\item a. Identify the center of the circumscribed circle, the centroid and the orthocenter of triangle ABG (no justification is expected).\\
b. Verify by calculation that these three points are indeed collinear.
\end{enumerate}