The exercises in the image involve proving certain properties in triangles and parallelograms based on vector relationships. Let's go through Exercise 12 and Exercise 13 one by one.

Exercise 12

We have triangle $ABC$ and a point $D$ such that: $\overrightarrow{CD} = \overrightarrow{AC} + 2 \overrightarrow{BC}$

1. Define $D_1$ and $D_2$ as projections of $D$ on $AB$ and $AC$, respectively.

   • a) Show that $\overrightarrow{AD_1} = -2 \overrightarrow{AB}$
   • b) Show that $\overrightarrow{AD_2} = 4 \overrightarrow{AC}$

2. Line $(CD)$ intersects lines $(D, D_2)$ and $(AB)$ at points $M$ and $N$, respectively.

   • a) Draw a figure to represent this setup.
   • b) Show that $\frac{AN}{AB} = \frac{2}{3}$
   • c) Determine the value of $\frac{MN}{MD}$.

Exercise 13

We have a parallelogram $ABCD$ with center $O$.

1. Let $J$ be a point such that $\overrightarrow{AJ} = \frac{2}{3} \overrightarrow{AC}$ and $E$ is the projection of $J$ on $BC$ parallel to $AB$.
   • a) Show that $\overrightarrow{CE} = \frac{1}{3} \overrightarrow{CB}$
   • b) Show that $\overrightarrow{JE} = \frac{1}{3} \overrightarrow{AB}$

Would you like detailed steps on how to solve these parts?

Additional Related Questions:

1. What properties of vectors are useful in proving segment ratios in triangles?
2. How can projection concepts help in simplifying vector proofs?
3. What role does the parallelogram law play in vector addition for quadrilaterals?
4. How can you generalize these proofs for other geometric shapes?
5. How do the concepts of projection and intersection relate in vector geometry?

Tip:

Remember that vector addition and scalar multiplication properties can simplify many geometric proofs by allowing you to decompose vectors along defined directions.