## circumcenter

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

The circumcenter is a point that is equidistant from the vertices of a triangle. In other words, it is the center of the circle that passes through all three vertices of the triangle.

To find the circumcenter of a triangle, you can use several methods. One method is to find the intersection point of the perpendicular bisectors of the triangle’s sides. The perpendicular bisector of a side is a line that is perpendicular to the side and passes through its midpoint.

Here are the steps to find the circumcenter using the perpendicular bisector method:

1. Take any two sides of the triangle and find their midpoints.

2. Find the slopes of the two sides.

3. Find the negative reciprocal of each slope to obtain the slopes of the perpendicular bisectors.

4. Use the midpoints and the slopes of the perpendicular bisectors to find the equations of those bisectors.

5. Find the intersection point of the two bisectors. This point will be the circumcenter of the triangle.

Another method to find the circumcenter is by using the coordinates of the triangle’s vertices. Let’s say the vertices of the triangle are (x1, y1), (x2, y2), and (x3, y3). You can use the following formulas to find the circumcenter coordinates (x, y):

x = (x1^2 + y1^2)(y2 – y3) + (x2^2 + y2^2)(y3 – y1) + (x3^2 + y3^2)(y1 – y2) / (2[(x1)(y2 – y3) + (x2)(y3 – y1) + (x3)(y1 – y2)])

y = (x1^2 + y1^2)(x3 – x2) + (x2^2 + y2^2)(x1 – x3) + (x3^2 + y3^2)(x2 – x1) / (2[(x1)(y2 – y3) + (x2)(y3 – y1) + (x3)(y1 – y2)])

These formulas can be derived using the fact that the circumcenter is equidistant from the three vertices.

Remember that the circumcenter is not always inside the triangle. It can also be outside or on the triangle, depending on the shape of the triangle.

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