Find the centroid of the triangle whose orthocentre is (-3,5) and circumcentre is (6,2).

Answer

To find the centroid of the triangle given its orthocenter and circumcenter, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: The centroid (G), orthocenter (H), and circumcenter (O) of a triangle are collinear. The centroid divides the line segment joining the orthocenter and circumcenter in the ratio 1:2.

2. **Identify the Coordinates**:
- Orthocenter (H) = (-3, 5)
- Circumcenter (O) = (6, 2)

3. **Apply the Section Formula**: The section formula states that if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of P are given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, G divides OH in the ratio 1:2, so m = 1 and n = 2.

4. **Calculate the x-coordinate of the Centroid (G)**:
\[ x_G = \frac{1 \cdot x_O + 2 \cdot x_H}{1 + 2} = \frac{1 \cdot 6 + 2 \cdot (-3)}{3} = \frac{6 - 6}{3} = \frac{0}{3} = 0 \]

5. **Calculate the y-coordinate of the Centroid (G)**:
\[ y_G = \frac{1 \cdot y_O + 2 \cdot y_H}{1 + 2} = \frac{1 \cdot 2 + 2 \cdot 5}{3} = \frac{2 + 10}{3} = \frac{12}{3} = 4 \]

6. **Final Coordinates of the Centroid**:
Therefore, the coordinates of the centroid G are (0, 4).

### Final Answer:
The centroid of the triangle is at the point (0, 4).

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