Tính AB BC AO

It looks like you have a geometry problem involving vectors and points in a triangle. Let's break down the steps to solve the questions presented:

### Given Information:
- **Points**: \( A(2, 9) \) and \( B(3, 5) \)
- Also references to points \( C \) and vector operations.

### Required Calculations:
1. **Vectors \( \vec{AB}, \vec{BC}, \vec{AC} \)**:
- Calculate \( \vec{AB} = B - A \)
- Find \( \vec{BC} \) and \( \vec{AC} \) similarly.

2. **Find the Centroid \( G \)**:
- The centroid \( G \) of triangle \( ABC \) is given by:
\[
G = \left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} \right)
\]

3. **Cross Product \(|AB \times BC|\)**:
- This can be calculated using the determinant method if you convert your vectors to a suitable form.

4. **Calculate Spring Values**:
- Use vector relationships to compute expressions involving coefficients you provided, like \( 2\vec{AB} - 3\vec{BC} \) and other operations as stated.

5. **Finding Point \( D \)**:
- The equations related to the point \( D \) and relationships between \( \vec{AB} \) and \( \vec{DC} \).

6. **Angle Between Vectors**:
- Use the dot product formula:
\[
\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta)
\]

If you need specific elements worked out (like specific vector calculations), let me know!