If the line joining the incentre to the centroid of a triangle ABC is parallel to the side BC. Which of the following are correct ?

Options

A. `2b=a+c`
B. `2a=b+c`
C. `"cot"(A)/(2)"cot"(C )/(2)=3 `
D. `"cot"(B)/(2) "cot"(C )/(2)=3 `

Answer

To solve the problem, we need to analyze the given condition that the line joining the incenter (I) to the centroid (G) of triangle ABC is parallel to side BC.

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
- We know that if the line IG is parallel to BC, then the slopes of IG and BC must be equal.

2. **Coordinates of Points**:
- Let the coordinates of the vertices of triangle ABC be:
- A = (x1, y1)
- B = (x2, y2)
- C = (x3, y3)

3. **Finding the Incenter (I)**:
- The coordinates of the incenter I can be calculated using the formula:
\[ I = \left( \frac{Ax_1 + Bx_2 + Cx_3}{A + B + C}, \frac{Ay_1 + By_2 + Cy_3}{A + B + C} \right) \]
- Here, A, B, and C are the lengths of the sides opposite to vertices A, B, and C respectively.

4. **Finding the Centroid (G)**:
- The coordinates of the centroid G are given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

5. **Finding the Slopes**:
- The slope of line BC is:
\[ \text{slope of BC} = \frac{y_2 - y_3}{x_2 - x_3} \]
- The slope of line IG is:
\[ \text{slope of IG} = \frac{\frac{Ay_1 + By_2 + Cy_3}{A + B + C} - \frac{y_1 + y_2 + y_3}{3}}{\frac{Ax_1 + Bx_2 + Cx_3}{A + B + C} - \frac{x_1 + x_2 + x_3}{3}} \]

6. **Setting the Slopes Equal**:
- Since IG is parallel to BC, we set the slopes equal:
\[ \frac{y_2 - y_3}{x_2 - x_3} = \frac{\frac{Ay_1 + By_2 + Cy_3}{A + B + C} - \frac{y_1 + y_2 + y_3}{3}}{\frac{Ax_1 + Bx_2 + Cx_3}{A + B + C} - \frac{x_1 + x_2 + x_3}{3}} \]

7. **Simplifying the Equation**:
- From the equality of slopes, we can derive relationships between the sides of the triangle. After simplification, we find:
\[ 2a = b + c \]
- This implies that the triangle is isosceles with sides b and c being equal to a.

8. **Verifying Other Conditions**:
- We can also check other conditions given in the options:
- \( a + b = c \) (False)
- \( a + c = b \) (False)
- \( \cot \frac{A}{2} \cot \frac{C}{2} = 3 \cot \frac{B}{2} \) (True)

### Conclusion:
Based on the analysis, the correct relationships derived from the conditions are:
- \( 2a = b + c \) is true.
- The other options provided are false.