circumcenter Chapter 6 (p. 300)
The point of concurrency of the three perpendicular bisectors of a triangle
The circumcenter is the point where the perpendicular bisectors of a triangle intersect. It is the center point of the circumcircle, which is the circle that passes through all three vertices of the triangle. The circumcenter is equidistant from the three vertices, meaning that the distance between the circumcenter and each vertex is the same.
To find the circumcenter of a triangle, we need to first draw the perpendicular bisectors of two sides of the triangle. We can then find the point where these two perpendicular bisectors intersect, which will be the circumcenter.
Alternatively, we can use the properties of the circumcenter to find its coordinates if we know the coordinates of the three vertices of the triangle. The circumcenter will be the intersection of the three perpendicular bisectors of the sides of the triangle, so we can find the equations of these bisectors using the midpoint formula and the slope formula. Solving for the intersection point will give us the coordinates of the circumcenter.
The circumcenter is an important point in a triangle as it provides information about the shape and size of the circumcircle. It can also be used in various geometric constructions and proofs.
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