Find the co-ordinates of the circumecentre of the triangle ABC, whose vertices A, B and C are (4, 6), (0,4) and (6,2) respectively.

Answer

To find the coordinates of the circumcenter of triangle ABC with vertices A(4, 6), B(0, 4), and C(6, 2), we will follow these steps:

### Step 1: Understand the circumcenter
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is equidistant from all three vertices of the triangle.

### Step 2: Find the midpoints of two sides
We can choose sides AB and AC to find their midpoints.

- **Midpoint of AB**:
\[ M_{AB} = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) = \left( \frac{4 + 0}{2}, \frac{6 + 4}{2} \right) = (2, 5) \]

- **Midpoint of AC**:
\[ M_{AC} = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{4 + 6}{2}, \frac{6 + 2}{2} \right) = (5, 4) \]

### Step 3: Find the slopes of the sides
Next, we calculate the slopes of sides AB and AC.

- **Slope of AB**:
\[ m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{4 - 6}{0 - 4} = \frac{-2}{-4} = \frac{1}{2} \]

- **Slope of AC**:
\[ m_{AC} = \frac{y_C - y_A}{x_C - x_A} = \frac{2 - 6}{6 - 4} = \frac{-4}{2} = -2 \]

### Step 4: Find the slopes of the perpendicular bisectors
The slopes of the perpendicular bisectors are the negative reciprocals of the slopes of the sides.

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

- **Slope of the perpendicular bisector of AC**:
\[ m_{PB_{AC}} = -\frac{1}{m_{AC}} = \frac{1}{2} \]

### Step 5: Write the equations of the perpendicular bisectors
Using the point-slope form of the line equation \(y - y_1 = m(x - x_1)\):

- **Equation of the perpendicular bisector of AB** (through M_{AB}(2, 5)):
\[ y - 5 = -2(x - 2) \implies y - 5 = -2x + 4 \implies y = -2x + 9 \]

- **Equation of the perpendicular bisector of AC** (through M_{AC}(5, 4)):
\[ y - 4 = \frac{1}{2}(x - 5) \implies y - 4 = \frac{1}{2}x - \frac{5}{2} \implies y = \frac{1}{2}x + \frac{3}{2} \]

### Step 6: Solve the equations simultaneously
To find the circumcenter, we need to solve the two equations:
1. \(y = -2x + 9\)
2. \(y = \frac{1}{2}x + \frac{3}{2}\)

Setting them equal to each other:
\[ -2x + 9 = \frac{1}{2}x + \frac{3}{2} \]

### Step 7: Solve for x
Multiply through by 2 to eliminate the fraction:
\[ -4x + 18 = x + 3 \]
Combine like terms:
\[ -4x - x = 3 - 18 \implies -5x = -15 \implies x = 3 \]

### Step 8: Solve for y
Substituting \(x = 3\) back into one of the equations:
\[ y = -2(3) + 9 = -6 + 9 = 3 \]

### Final Answer
The coordinates of the circumcenter of triangle ABC are \((3, 3)\).

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