The vertices of a triangle are (1,2) (2,1) and `{1/2(3+sqrt(3)),1/2(3+sqrt(3))}`. The dstnace between its orthocentre and circumcentre is

Options

A. `(3+sqrt(3))`
B. 2
C. `sqrt(2)`
D. `0`

Answer

To find the distance between the orthocenter and circumcenter of the triangle with vertices \( A(1, 2) \), \( B(2, 1) \), and \( C\left(\frac{1}{2}(3+\sqrt{3}), \frac{1}{2}(3+\sqrt{3})\right) \), we will follow these steps:

### Step 1: Find the coordinates of the vertices
The vertices of the triangle are:
- \( A(1, 2) \)
- \( B(2, 1) \)
- \( C\left(\frac{1}{2}(3+\sqrt{3}), \frac{1}{2}(3+\sqrt{3})\right) \)

### Step 2: Calculate the lengths of the sides of the triangle
Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \):

1. **Length of side \( AB \)**:
\[ AB = \sqrt{(2 - 1)^2 + (1 - 2)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

2. **Length of side \( BC \)**:
\[ BC = \sqrt{\left(\frac{1}{2}(3+\sqrt{3}) - 2\right)^2 + \left(\frac{1}{2}(3+\sqrt{3}) - 1\right)^2} \]
Simplifying:
\[ = \sqrt{\left(\frac{3+\sqrt{3}}{2} - \frac{4}{2}\right)^2 + \left(\frac{3+\sqrt{3}}{2} - \frac{2}{2}\right)^2} \]
\[ = \sqrt{\left(\frac{-1+\sqrt{3}}{2}\right)^2 + \left(\frac{1+\sqrt{3}}{2}\right)^2} \]
\[ = \sqrt{\frac{(1-\sqrt{3})^2 + (1+\sqrt{3})^2}{4}} = \sqrt{\frac{1 - 2\sqrt{3} + 3 + 1 + 2\sqrt{3} + 3}{4}} = \sqrt{\frac{8}{4}} = \sqrt{2} \]

3. **Length of side \( AC \)**:
\[ AC = \sqrt{\left(\frac{1}{2}(3+\sqrt{3}) - 1\right)^2 + \left(\frac{1}{2}(3+\sqrt{3}) - 2\right)^2} \]
Simplifying:
\[ = \sqrt{\left(\frac{3+\sqrt{3}}{2} - 1\right)^2 + \left(\frac{3+\sqrt{3}}{2} - 2\right)^2} \]
\[ = \sqrt{\left(\frac{1+\sqrt{3}}{2}\right)^2 + \left(\frac{-1+\sqrt{3}}{2}\right)^2} \]
\[ = \sqrt{\frac{(1+\sqrt{3})^2 + (-1+\sqrt{3})^2}{4}} = \sqrt{\frac{1 + 2\sqrt{3} + 3 + 1 - 2\sqrt{3} + 3}{4}} = \sqrt{\frac{8}{4}} = \sqrt{2} \]

### Step 3: Find the circumcenter
The circumcenter \( O \) of a triangle can be found using the perpendicular bisectors of the sides. Since all sides are equal, the circumcenter will be at the centroid \( G \) of the triangle.

The coordinates of the centroid \( G \) are given by:
\[ G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
Calculating:
\[ G = \left(\frac{1 + 2 + \frac{3+\sqrt{3}}{2}}{3}, \frac{2 + 1 + \frac{3+\sqrt{3}}{2}}{3}\right) \]
\[ = \left(\frac{3 + \frac{3+\sqrt{3}}{2}}{3}, \frac{3 + \frac{3+\sqrt{3}}{2}}{3}\right) = \left(\frac{6 + \sqrt{3}}{6}, \frac{6 + \sqrt{3}}{6}\right) \]

### Step 4: Find the orthocenter
For an equilateral triangle, the orthocenter \( H \) coincides with the centroid. Thus, \( H = G \).

### Step 5: Calculate the distance between the orthocenter and circumcenter
Since \( H \) and \( O \) are the same point, the distance \( d \) between them is:
\[ d = 0 \]

### Final Answer
The distance between the orthocenter and circumcenter is \( 0 \).

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