1. **Problem Statement:** Given parallelogram ABCD, point P lies on AD such that $AP=\frac{1}{3}AD$, and point Q lies on BC such that $CQ=\frac{1}{3}BC$. We need to analyze or solve the problem involving these points. 2. **Understanding the Parallelogram and Points:** In a parallelogram, opposite sides are equal and parallel. Here, AD and BC are opposite sides. 3. **Position Vectors Setup:** Let’s assign position vectors to points for clarity: - Let $\vec{A}=\vec{0}$ (origin for convenience). - Let $\vec{D}=\vec{d}$. - Since ABCD is a parallelogram, $\vec{B}=\vec{b}$ and $\vec{C}=\vec{b}+\vec{d}$. 4. **Coordinates of Points P and Q:** - Point P lies on AD such that $AP=\frac{1}{3}AD$, so $$\vec{P} = \vec{A} + \frac{1}{3}(\vec{D} - \vec{A}) = \frac{1}{3}\vec{d}.$$ - Point Q lies on BC such that $CQ=\frac{1}{3}BC$, so $$\vec{Q} = \vec{C} - \frac{1}{3}(\vec{C} - \vec{B}) = (\vec{b} + \vec{d}) - \frac{1}{3}\vec{d} = \vec{b} + \frac{2}{3}\vec{d}.$$ 5. **Interpretation:** - $\vec{P}$ divides AD in ratio 1:2 starting from A. - $\vec{Q}$ divides BC in ratio 2:1 starting from B. 6. **Further Analysis (Example: Find Vector PQ):** $$\vec{PQ} = \vec{Q} - \vec{P} = \left(\vec{b} + \frac{2}{3}\vec{d}\right) - \frac{1}{3}\vec{d} = \vec{b} + \frac{1}{3}\vec{d}.$$ 7. **Summary:** We have expressed points P and Q in terms of vectors $\vec{b}$ and $\vec{d}$, which represent sides AB and AD respectively. This vector approach helps solve related problems such as finding lengths, midpoints, or proving properties. **Final answer:** $$\vec{P} = \frac{1}{3}\vec{d}, \quad \vec{Q} = \vec{b} + \frac{2}{3}\vec{d}.$$