In the following triangle \(▵ABC\),

\(AK\), \(BL\) and \(CJ\) are the **perpendicular bisectors** of the triangle.

They intersect at the point \(O\),
which is the **circumcenter** of \(△ABC\).

The circumcenter point of a triangle is equidistant
from all the vertices of that triangle.

The **circumcircle** of a triangle is a circle that passes through
all the three vertices of that triangle.

The center of a circumcircle is known as the **circumcenter**,
while radius of the circumcircle is called the **circumradius**.

A circumcircle is also known as the **circumscribed circle**.

A perpendicular bisector is a line that bisects another line in two equal parts and makes an angle of \(90°\) at the point of intersection. It is a perpendicular on the midpoint of another line.

In the above, \(N\) is the midpoint of \(OP\).

So, \(MN\) has bisected \(OP\) in two equal parts.

Again, \(MN\) is also perpendicular to \(OP\).

So, \(MN\) is the

- ✓ Circumcenter is the point of intersection of the perpendicular bisectors.
- ✓ It is the centre of the circumcircle of the triangle.
- ✓ All the vertices of the triangle are equidistant from the circumcenter.
- ✓ For an acute triangle, it lies
**inside the triangle.** - ✓ For an obtuse triangle, it lies
**outside the triangle.** - ✓ For a right-angled triangle, it lies
**on the midpoint of the hypotenuse.** - ✓ In an equilateral triangle, the centroid, the orthocenter,
- the
**circumcenter**and the incenter coincide on the same point.

\(OJ\), \(OK\) and \(OL\) are the **perpendicular bisectors**.

\(O\) is the **circumcenter** of the triangle.

For an acute triangle, the circumcenter lies **inside the triangle**.

\(OA\), \(OB\) and \(OC\) are the **circumradii**.

The circle is the **circumcircle** of the triangle.

Note that the triangle is isosceles too.

For an obtuse triangle, the circumcenter lies **outside the triangle**.

For an right-angled triangle, the circumcenter lies
on the **midpoint of the hypotenuse** of the triangle,
and the hypotenuse becomes the diameter of the circumcircle.

If coordinates of the vertices \(A\), \(B\) and \(C\) are
\( (-18, 9) \), \( (11, 9) \) and \( (-9, 33) \) respectively,
then the coordinates of the circumcenter
\(O(x, y)\) can be calculated with the following steps:

Coordinates of \(O = (x, y) \)

Using distance formula \(\; \sqrt{(y_2-y_1)^2 + (x_2-x_1)^2} \)** ,**

Distance O to A = \(\; OA = \sqrt{(y - 9)^2 + (x - (-18))^2} \)

Distance O to B = \(\; OB = \sqrt{(y - 9)^2 + (x - 11)^2} \)

Distance O to C = \(\; OC = \sqrt{(y - 33)^2 + (x - (-9))^2} \)

Because \(OA\), \(OB\) and \(OC\) are radii of the same circle,
we can solve for \(x\) and \(y\) from the above information.

\begin{align} & OA = OB \quad\color{gray}\text{[Being the radii of same circle]} \\\\ \Rightarrow\; & \sqrt{(y - 9)^2 + (x - (-18))^2} = \sqrt{(y - 9)^2 + (x - 11)^2} \\\\ \Rightarrow\; & (y - 9)^2 + (x + 18)^2 = (y - 9)^2 + (x - 11)^2 \\\\ \Rightarrow\; & (x + 18)^2 = (x - 11)^2 \\\\ \Rightarrow\; & x^2 + 2\cdot x\cdot 18 + 18^2 = x^2 - 2\cdot x\cdot 11 + (11)^2 \\\\ \Rightarrow\; & x^2 + 36x + 324 = x^2 - 22x + 121 \\\\ \Rightarrow\; & 36x + 22x = 121-324 \\\\ \Rightarrow\; & 48x = -203 \\\\ \therefore\; & x = -3.5 \\\\ \end{align}

Solving for \(x\) and \(y\) gives us the coordinates of the circumcenter at \(O(-3.5, 17.25)\)

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