Find the circumcentre of the triangle whose vertices are (2, 2), (4, 2) and (0, 4).

Answer

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

### Step 1: Identify the midpoints of two sides of the triangle.
- **Midpoint of AB**:
\[ M_{AB} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 4}{2}, \frac{2 + 2}{2} \right) = (3, 2) \]
- **Midpoint of AC**:
\[ M_{AC} = \left( \frac{2 + 0}{2}, \frac{2 + 4}{2} \right) = (1, 3) \]

### Step 2: Find the slopes of the sides AB and AC.
- **Slope of AB (M_{AB})**:
\[ M_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 2}{4 - 2} = 0 \]
- **Slope of AC (M_{AC})**:
\[ M_{AC} = \frac{4 - 2}{0 - 2} = \frac{2}{-2} = -1 \]

### Step 3: Find the slopes of the perpendicular bisectors.
- The slope of the perpendicular bisector of AB (M_{pAB}) is the negative reciprocal of M_{AB}:
\[ M_{pAB} = \text{undefined} \quad (\text{since } M_{AB} = 0) \]
This means the perpendicular bisector is a vertical line through the midpoint (3, 2), which is \( x = 3 \).

- The slope of the perpendicular bisector of AC (M_{pAC}) is the negative reciprocal of M_{AC}:
\[ M_{pAC} = 1 \]
The equation of the perpendicular bisector can be written using point-slope form:
\[ y - 3 = 1(x - 1) \implies y = x + 2 \]

### Step 4: Find the intersection of the two perpendicular bisectors.
We have the equations:
1. \( x = 3 \) (perpendicular bisector of AB)
2. \( y = x + 2 \) (perpendicular bisector of AC)

Substituting \( x = 3 \) into the second equation:
\[ y = 3 + 2 = 5 \]

### Step 5: Conclusion
The circumcenter of the triangle is at the point \( (3, 5) \).

### Summary
The circumcenter of the triangle with vertices (2, 2), (4, 2), and (0, 4) is \( (3, 5) \).

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