For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. In space with an orthonormal coordinate system, consider the following points: $$\mathrm{A}(2;0;0), \quad \mathrm{B}(0;4;3), \quad \mathrm{C}(4;4;1), \quad \mathrm{D}(0;0;4) \text{ and } \mathrm{H}(-1;1;2)$$ Statement 1: the points A, C and D define a plane $\mathscr{P}$ with equation $8x - 5y + 4z - 16 = 0$. Statement 2: the points A, B, C and D are coplanar. Statement 3: the lines $(\mathrm{AC})$ and $(\mathrm{BH})$ are secant. It is admitted that the plane (ABC) has the Cartesian equation $x - y + 2z - 2 = 0$. Statement 4: the point H is the orthogonal projection of point D onto the plane (ABC).
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
In space with an orthonormal coordinate system, consider the following points:
$$\mathrm{A}(2;0;0), \quad \mathrm{B}(0;4;3), \quad \mathrm{C}(4;4;1), \quad \mathrm{D}(0;0;4) \text{ and } \mathrm{H}(-1;1;2)$$
\textbf{Statement 1:} the points A, C and D define a plane $\mathscr{P}$ with equation $8x - 5y + 4z - 16 = 0$.\\
\textbf{Statement 2:} the points A, B, C and D are coplanar.\\
\textbf{Statement 3:} the lines $(\mathrm{AC})$ and $(\mathrm{BH})$ are secant.\\
It is admitted that the plane (ABC) has the Cartesian equation $x - y + 2z - 2 = 0$.\\
\textbf{Statement 4:} the point H is the orthogonal projection of point D onto the plane (ABC).