This exercise involves vector analysis within a triangle and uses projections to determine relationships between segments. Let’s go through the problem step-by-step.

The problem specifies:

1. $ABC$ is a triangle, and $D$ is a point such that: $\vec{CD} = \vec{AC} + 2 \vec{BC}$
2. Points $D_1$ and $D_2$ are projections of $D$ on lines $(AB)$ and $(AC)$, respectively, with the following instructions:

Part 1

1. a) Show that $\vec{AD_1} = -2 \vec{AB}$.
2. b) Show that $\vec{AD_2} = 4 \vec{AC}$.

Part 2

1. Construct a figure based on these points.
2. a) Show that $\frac{\vec{AN}}{\vec{AB}} = \frac{2}{3}$.
3. b) Find the value of $\frac{\vec{MN}}{\vec{MD}}$.

Solution Steps

To proceed, let's first clarify a few steps and assumptions that might be useful:

1. Vector Setup:

   • Since $\vec{CD} = \vec{AC} + 2 \vec{BC}$, you can use this to position $D$ in terms of known vectors in the triangle.
   • Using the projection properties, apply geometric constraints on $D_1$ and $D_2$ to analyze the vectors along $(AB)$ and $(AC)$.

2. Figure Construction:

   • Construct the triangle $ABC$, place $D$ relative to points $A$, $B$, and $C$, and mark points $D_1$, $D_2$, $M$, and $N$ accordingly.

Would you like a detailed step-by-step breakdown of each part, or is there a specific part you need more help with?

Here are five related questions to deepen understanding:

1. What is the significance of vector projections in geometry?
2. How does the position of point $D$ influence the projections onto $(AB)$ and $(AC)$?
3. What role does the triangle’s orientation play in the interpretation of vector results?
4. How does the vector form of $CD$ help determine the location of point $D$ within the triangle?
5. What mathematical properties are used to confirm relationships like $\vec{AD_1} = -2 \vec{AB}$?

Tip: When working with vector projections, using coordinate geometry can simplify calculations by providing concrete representations of points.