In `Delta ABC` if the orthocentre is `(1,2)` and the circumcenter is `(0,0)` then centroid of `Delta ABC` is.

Options

A. `(1//2,2//3)`
B. `(1//3,2//3)`
C. `(2//3,1)`
D. none of these

Explanation

To find the coordinates of the centroid \( G \) of triangle \( ABC \) given the orthocenter \( H(1, 2) \) and the circumcenter \( O(0, 0) \), we can use the property that the centroid divides the line segment joining the orthocenter and circumcenter in the ratio \( 1:2 \).

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Orthocenter \( H = (1, 2) \)
- Circumcenter \( O = (0, 0) \)

2. **Use the Section Formula:**
The centroid \( G \) divides the line segment \( HO \) in the ratio \( 1:2 \). According to the section formula, if a point \( G \) divides the line segment joining points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \), then the coordinates of \( G \) can be calculated as:
\[ G\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

3. **Apply the Section Formula:**
Here, \( m = 1 \), \( n = 2 \), \( (x_1, y_1) = (1, 2) \), and \( (x_2, y_2) = (0, 0) \).

- For the x-coordinate:
\[ G_x = \frac{1 \cdot 0 + 2 \cdot 1}{1 + 2} = \frac{0 + 2}{3} = \frac{2}{3} \]

- For the y-coordinate:
\[ G_y = \frac{1 \cdot 0 + 2 \cdot 2}{1 + 2} = \frac{0 + 4}{3} = \frac{4}{3} \]

4. **Final Coordinates of the Centroid:**
Thus, the coordinates of the centroid \( G \) are:
\[ G\left( \frac{2}{3}, \frac{4}{3} \right) \]

### Conclusion:
The coordinates of the centroid \( G \) of triangle \( ABC \) are \( \left( \frac{2}{3}, \frac{4}{3} \right) \).

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