The vertices of a triangle are `(4,-3)`, `(-2,1)` and `(2,3)` . Find the co-ordinates of the circumcentre of the triangle.
[Circumcentre is the point of concurrence of the right-bisectors of the sides of a triangle]

Answer

To find the coordinates of the circumcenter of the triangle with vertices at \( A(4, -3) \), \( B(-2, 1) \), and \( C(2, 3) \), we will follow these steps:

### Step 1: Find the midpoints of two sides of the triangle.

**Midpoint of AB:**
\[ D = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{4 + (-2)}{2}, \frac{-3 + 1}{2} \right) = \left( \frac{2}{2}, \frac{-2}{2} \right) = (1, -1) \]

**Midpoint of AC:**
\[ E = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{4 + 2}{2}, \frac{-3 + 3}{2} \right) = \left( \frac{6}{2}, \frac{0}{2} \right) = (3, 0) \]

### Step 2: Calculate the slopes of sides AB and AC.

**Slope of AB:**
\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-3)}{-2 - 4} = \frac{4}{-6} = -\frac{2}{3} \]

**Slope of AC:**
\[ m_{AC} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - (-3)}{2 - 4} = \frac{6}{-2} = -3 \]

### Step 3: Find the slopes of the perpendicular bisectors.

**Slope of the perpendicular bisector of AB:**
\[ m_{D} = -\frac{1}{m_{AB}} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2} \]

**Slope of the perpendicular bisector of AC:**
\[ m_{E} = -\frac{1}{m_{AC}} = -\frac{1}{-3} = \frac{1}{3} \]

### Step 4: Write the equations of the perpendicular bisectors.

**Equation of the perpendicular bisector of AB (using point D(1, -1)):**
\[ y - y_1 = m(x - x_1) \implies y + 1 = \frac{3}{2}(x - 1) \]
\[ y + 1 = \frac{3}{2}x - \frac{3}{2} \implies y = \frac{3}{2}x - \frac{5}{2} \quad \text{(Equation 1)} \]

**Equation of the perpendicular bisector of AC (using point E(3, 0)):**
\[ y - y_1 = m(x - x_1) \implies y - 0 = \frac{1}{3}(x - 3) \]
\[ y = \frac{1}{3}x - 1 \quad \text{(Equation 2)} \]

### Step 5: Solve the equations simultaneously to find the circumcenter.

Set Equation 1 equal to Equation 2:
\[ \frac{3}{2}x - \frac{5}{2} = \frac{1}{3}x - 1 \]

To eliminate the fractions, multiply through by 6:
\[ 6\left(\frac{3}{2}x - \frac{5}{2}\right) = 6\left(\frac{1}{3}x - 1\right) \]
\[ 9x - 15 = 2x - 6 \]
\[ 9x - 2x = 15 - 6 \]
\[ 7x = 9 \implies x = \frac{9}{7} \]

Now substitute \( x \) back into one of the equations to find \( y \):
Using Equation 2:
\[ y = \frac{1}{3}\left(\frac{9}{7}\right) - 1 = \frac{3}{7} - 1 = \frac{3}{7} - \frac{7}{7} = -\frac{4}{7} \]

### Final Answer:
The coordinates of the circumcenter are:
\[ \left( \frac{9}{7}, -\frac{4}{7} \right) \]