The problem presented involves a triangle $ABC$ and points $I$ and $J$ with specific geometric relationships. Let's break it down step by step:

Problem Analysis:

• Given Information:
  • Triangle $ABC$.
  • Points $I$ and $J$ are two points in the plane such that: $\overrightarrow{AI} = \frac{1}{3} \overrightarrow{AB}.$
  • Point $J$ is the projection of point $I$ onto line $BC$ along a line parallel to $AC$, and point $Q$ is the projection of $J$ onto $AC$ along a line parallel to $AB$.

Tasks:

1. Construction of the Figure:

   • You are asked to construct a diagram based on the given points, projections, and triangle.

2. Show that $\overrightarrow{CJ} = \frac{1}{3} \overrightarrow{CB}$:

   • This likely involves using properties of projections and vectors.

3. Show that $\overrightarrow{AC} = 3 \overrightarrow{QC}$:

   • This will involve analyzing the projection relationships between the points $J$, $Q$, and $AC$.

Approach:

1. Construction of the figure:

   • Draw triangle $ABC$.
   • Locate point $I$ along the segment $AB$, specifically $1/3$ of the way from $A$ to $B$.
   • From $I$, project point $J$ onto the line $BC$ parallel to $AC$.
   • Finally, project point $J$ onto line $AC$ parallel to $AB$, creating point $Q$.

2. Show $\overrightarrow{CJ} = \frac{1}{3} \overrightarrow{CB}$:

   • Using similar triangles or projection properties, the relationship between the vector from $C$ to $J$ and the full length of $CB$ can be demonstrated as proportional.

3. Show $\overrightarrow{AC} = 3 \overrightarrow{QC}$:

   • Again, using projection and similar triangle arguments, the distance $QC$ can be shown as a fraction of $AC$, leading to the result $AC = 3QC$.

Would you like further details on each step? Here are five related questions to deepen your understanding:

1. How do vector projections work in geometry?
2. What properties of parallel projections can be used to prove the second result?
3. How can similar triangles be used to relate the vectors $CJ$ and $CB$?
4. How do you construct the figure geometrically from the given vector relationships?
5. How can we use ratios of distances to solve projection problems?

Tip: Always start by drawing a clear diagram when tackling geometry problems, as this helps visualize the relationships between the points and lines.