To solve the problem step by step, we will follow the given information and apply the concept of section formula and ratios.
### Step 1: Understand the given points and ratios
We have three vertices of a triangle \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \). The points \( P \), \( Q \), and \( R \) divide the sides of the triangle in specified ratios.
### Step 2: Find the coordinates of point P
Point \( P \) divides segment \( BC \) in the ratio \( 2:1 \). Using the section formula, the coordinates of point \( P \) can be calculated as follows:
\[
P\left(\frac{2x_3 + 1x_2}{2 + 1}, \frac{2y_3 + 1y_2}{2 + 1}\right) = P\left(\frac{2x_3 + x_2}{3}, \frac{2y_3 + y_2}{3}\right)
\]
### Step 3: Find the coordinates of point Q
Point \( Q \) divides segment \( CA \) in the ratio \( 1:3 \). Using the section formula again, the coordinates of point \( Q \) can be calculated as follows:
\[
Q\left(\frac{1x_1 + 3x_3}{1 + 3}, \frac{1y_1 + 3y_3}{1 + 3}\right) = Q\left(\frac{x_1 + 3x_3}{4}, \frac{y_1 + 3y_3}{4}\right)
\]
### Step 4: Set up the relationship for point R
Since \( P \), \( Q \), and \( R \) are points dividing the sides of triangle \( ABC \), we can use the property that the product of the ratios of the segments divided by these points is equal to -1:
\[
\frac{P}{BC} \cdot \frac{Q}{CA} \cdot \frac{R}{AB} = -1
\]
Substituting the known values:
\[
\frac{2}{1} \cdot \frac{1}{3} \cdot \frac{R}{AB} = -1
\]
### Step 5: Solve for R
Now we can simplify the equation:
\[
\frac{2}{1} \cdot \frac{1}{3} = \frac{2}{3}
\]
Thus, we have:
\[
\frac{2}{3} \cdot \frac{R}{AB} = -1
\]
To isolate \( R \):
\[
\frac{R}{AB} = -\frac{3}{2}
\]
This means that the ratio \( R:AB = -3:2 \).
### Step 6: Interpret the negative sign
The negative sign indicates that point \( R \) divides segment \( AB \) externally. Therefore, the ratio of division is \( 3:2 \) externally.
### Conclusion
Thus, point \( R \) divides segment \( AB \) in the ratio \( 3:2 \) externally.
