## circumcenter Chapter 6 (p. 300)

### In geometry, the circumcenter is a significant point associated with a triangle

In geometry, the circumcenter is a significant point associated with a triangle. It is the center of a circle that circumscribes or passes through all three vertices of the triangle. The circumcenter is denoted by the letter ‘O’.

To find the circumcenter of a triangle, we follow a specific method. Here are the steps:

1. Draw the triangle: Start by drawing the given triangle with its three sides and three vertices.

2. Find the perpendicular bisectors: A perpendicular bisector is a line that divides a line segment into two equal halves and is perpendicular to the line segment. To find the circumcenter, we need to find the perpendicular bisectors of at least two sides of the triangle.

3. Construct the perpendicular bisectors: Take any two sides of the triangle and construct their perpendicular bisectors. To do this, use a compass to measure the same distance from each endpoint of a side and draw arcs. Connect the two intersection points of these arcs to form the perpendicular bisector. Repeat this process for the other pair of sides.

4. Locate the circumcenter: The circumcenter of a triangle is the point where all the perpendicular bisectors intersect. This point represents the center of the circle that passes through all three vertices of the triangle.

5. Mark the circumcenter: Once you have determined the point of intersection of the perpendicular bisectors, mark it as ‘O’ to denote the circumcenter.

It is important to note that the circumcenter is equidistant from all three vertices of the triangle. This means that the distances between the circumcenter and each vertex are equal.

Knowing the circumcenter of a triangle can be useful when dealing with properties and measurements related to triangles, such as finding the radius of the circumcircle or determining angles and distances involving the circumcenter.

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