Circumcenter of a Triangle
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Examples, solutions, videos, worksheets, games and activities to help Geometry students learn how to construct the circumcenter of a triangle.
What is the circumcenter of a triangle?
The circumcenter of a triangle is one of the four classical triangle centers, along with the orthocenter, centroid, and incenter.
The point of concurrency of the three perpendicular bisectors of a triangle is the circumcenter. It is the center of the circle circumscribed about the triangle, making the circumcenter equidistant from the three vertices of the triangle. The circumcenter is not always within the triangle. In a coordinate plane, to find the circumcenter we first find the equation of two perpendicular bisectors of the sides and solve the system of equations.
The following diagrams show the circumcenters for an acute triangle, a right triangle, and an obtuse triangle. Scroll down the page for more examples and solutions on the circumcenters of triangles.
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What is a Perpendicular Bisector?
A perpendicular bisector of a line segment is a line that:
- Passes through the midpoint of the segment.
- Is perpendicular to the segment.
How to Find the Circumcenter
To locate the circumcenter of a triangle:
- Draw (or construct) a perpendicular bisector through at least two of the triangle’s sides.
- The point where these two (or three) perpendicular bisectors intersect is the circumcenter.
Key Properties and Significance
The circumcenter is significant because it is the center of the triangle’s circumscribed circle (or circumcircle).
- Equidistant from Vertices: The circumcenter is equidistant from all three vertices of the triangle. This distance is the circumradius (R) of the circumcircle. This means if you place the point of a compass on the circumcenter and open it to any vertex, you can draw a circle that passes through all three vertices of the triangle.
- Unique Circumcircle: Every triangle has exactly one circumcircle and, therefore, exactly one circumcenter.
- Position depends on the triangle type:
Acute Triangle: Inside the triangle.
Right Triangle: On the hypotenuse (midpoint of the hypotenuse).
Obtuse Triangle: Outside the triangle.
How to construct the circumcenter of an acute triangle, a right triangle and an obtuse angle?
Construct Circumcenter of an Acute Triangle
The circumcenter of an acute triangle is located inside the triangle.
Construct Circumcenter of a Right Triangle
The circumcenter of a right triangle is at the midpoint of the hypotenuse.
Construct Circumcenter of an Obtuse Triangle
The circumcenter of an obtuse triangle is outside the triangle opposite the obtuse angle.
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