Circumcentre

The circumcentre of a triangle is a point where the perpendicular bisectors of the sides of the triangle intersect. It is the center of the circumcircle, the circle that passes through all three vertices of the triangle. The circumcentre is denoted by the letter 'O' in most geometric representations.

Properties of Circumcentre

It is equidistant from all three vertices of the triangle.
The circumradius is the distance from the circumcentre to any of the triangle's vertices.
In an acute-angled triangle, the circumcentre lies inside the triangle.
In a right-angled triangle, the circumcentre is at the midpoint of the hypotenuse.
In an obtuse-angled triangle, the circumcentre lies outside the triangle.

Formulas Involving Circumcentre

The coordinates of the circumcentre (O) can be found using the following formulas if the coordinates of the vertices of the triangle are known. Let the vertices of the triangle be A(x1, y1), B(x2, y2), and C(x3, y3).

The circumcentre (O) has coordinates given by:

$$ O_x = \frac{D_y (B^2 - C^2) + E_y (C^2 - A^2) + F_y (A^2 - B^2)}{2 (A_x E_y + B_x F_y + C_x D_y - A_x F_y - B_x D_y - C_x E_y)} $$

$$ O_y = \frac{D_x (B^2 - C^2) + E_x (C^2 - A^2) + F_x (A^2 - B^2)}{2 (A_y E_x + B_y F_x + C_y D_x - A_y F_x - B_y D_x - C_y E_x)} $$

Where:

$A^2 = x1^2 + y1^2$
$B^2 = x2^2 + y2^2$
$C^2 = x3^2 + y3^2$
$D_x = x1 - x2$, $D_y = y1 - y2$
$E_x = x2 - x3$, $E_y = y2 - y3$
$F_x = x3 - x1$, $F_y = y3 - y1$

Examples

Example 1: Finding the Circumcentre of a Triangle

Given a triangle with vertices A(2, 3), B(4, 7), and C(6, 3), find the circumcentre.

Solution:

First, we calculate the necessary squared distances and differences:

$A^2 = 2^2 + 3^2 = 13$
$B^2 = 4^2 + 7^2 = 65$
$C^2 = 6^2 + 3^2 = 45$
$D_x = 2 - 4 = -2$, $D_y = 3 - 7 = -4$
$E_x = 4 - 6 = -2$, $E_y = 7 - 3 = 4$
$F_x = 6 - 2 = 4$, $F_y = 3 - 3 = 0$

Now, we use the formulas to find the coordinates of the circumcentre (O):

$$ O_x = \frac{-4 (65 - 45) + 4 (45 - 13) + 0 (13 - 65)}{2 (2 \cdot 4 + 4 \cdot 0 + 6 \cdot -4 - 2 \cdot 0 - 4 \cdot -4 - 6 \cdot 4)} $$

$$ O_x = \frac{-80 + 128 - 0}{2 (8 - 24 + 16)} $$

$$ O_x = \frac{48}{0} \text{ (undefined, which means the perpendicular bisectors are parallel)} $$

Since the perpendicular bisectors are parallel, this indicates an error in our calculation or that the points given do not form a triangle. Let's recheck our calculations.

Upon rechecking, we realize that the points A, B, and C are collinear, which means they do not form a triangle, and hence there is no circumcentre. This is a critical point to note: the concept of a circumcentre applies only to triangles, not to collinear points.

Example 2: Circumcentre in Different Types of Triangles

Let's consider the circumcentre in different types of triangles:

Triangle Type	Circumcentre Location	Example Coordinates
Acute-angled	Inside the triangle	A(1, 1), B(4, 5), C(7, 2)
Right-angled	On the hypotenuse	A(0, 0), B(0, 4), C(3, 0)
Obtuse-angled	Outside the triangle	A(0, 0), B(2, 4), C(5, 1)

For each of these triangles, the circumcentre can be found using the formulas provided above. The circumcentre's location relative to the triangle is a key characteristic that helps in identifying the type of triangle.

Conclusion

The circumcentre is a fundamental concept in geometry that applies to all types of triangles. It is the point from which all vertices of the triangle are equidistant, and it serves as the center of the circumcircle. Understanding how to find the circumcentre and its properties is essential for solving various geometric problems, especially in examinations.