How To Find The Circumcenter Of A Triangle: Step-By-Step Guide With Formulas And Examples

Circumcenter

the point of concurrency of the perpendicular bisectors of a triangle

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

To find the circumcenter of a triangle, you need to find the intersection point of the perpendicular bisectors of the sides of the triangle. 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 of a triangle:

1. Identify the coordinates of the three vertices of the triangle (A, B, and C).
2. Find the midpoint of each side of the triangle. For example, the midpoint of side AB is the point ( (xA + xB)/2, (yA + yB)/2).
3. Find the equation of the perpendicular bisector of each side. The equation of the perpendicular bisector of a line segment with endpoints ( x1, y1) and ( x2, y2) is the line y = mx + b, where m is the negative reciprocal of the slope, and b is the y-intercept that passes through the midpoint of the line segment.
4. Solve for the point of intersection of the three perpendicular bisectors. This point is the circumcenter.

Alternatively, you can also use the distance formula to find the distance between each pair of vertices, and then use the distance formula again to find the coordinates of the circumcenter. The coordinates of the circumcenter will be the intersection point of two perpendicular bisectors of any two sides of the triangle.

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