To solve the problem, we need to find the coordinates of point \( P(x, y) \) such that the ratio of the distances \( \frac{AP}{PB} = \frac{3}{2} \). The internal bisector of angle \( \angle APB \) will pass through a specific point based on this ratio.
### Step-by-Step Solution:
1. **Identify the Coordinates of Points A and B:**
- Point \( A(5, 2) \)
- Point \( B(10, 12) \)
2. **Set Up the Ratio:**
- Given \( \frac{AP}{PB} = \frac{3}{2} \), we can denote:
- \( AP = 3k \)
- \( PB = 2k \)
- Therefore, the total distance \( AB \) can be expressed as:
\[
AB = AP + PB = 3k + 2k = 5k
\]
3. **Use the Section Formula:**
- The coordinates of point \( P \) that divides the line segment \( AB \) in the ratio \( 3:2 \) can be found using the section formula:
\[
P\left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
\]
where \( m = 3 \), \( n = 2 \), \( A(x_1, y_1) = (5, 2) \), and \( B(x_2, y_2) = (10, 12) \).
4. **Calculate the Coordinates of Point P:**
- For the x-coordinate:
\[
x = \frac{3 \cdot 10 + 2 \cdot 5}{3 + 2} = \frac{30 + 10}{5} = \frac{40}{5} = 8
\]
- For the y-coordinate:
\[
y = \frac{3 \cdot 12 + 2 \cdot 2}{3 + 2} = \frac{36 + 4}{5} = \frac{40}{5} = 8
\]
- Thus, the coordinates of point \( P \) are \( P(8, 8) \).
5. **Determine the Internal Bisector:**
- The internal bisector of angle \( \angle APB \) will pass through point \( P \) and will also pass through the point that divides \( AB \) in the ratio \( AP:PB \).
6. **Conclusion:**
- Since we found that \( P(8, 8) \) is the point that satisfies the given condition, we can conclude that the internal bisector of \( \angle APB \) passes through the point \( (8, 8) \).
### Final Answer:
The internal bisector of \( \angle APB \) always passes through the point \( (8, 8) \).
