The circumcenter of a triangle possesses several key properties:
1. It is equidistant from the three vertices of the triangle. This means that the distance from the circumcenter to each vertex is the same.
2. The circumcenter lies on the perpendicular bisectors of the triangle's sides. Each side's perpendicular bisector passes through the circumcenter.
3. It is the center of the triangle's circumcircle, which is a circle that passes through all three vertices of the triangle.
4. The circumcenter is the point of intersection of the perpendicular bisectors of any two sides of the triangle.
5. It lies inside the triangle if and only if the triangle is acute, outside of it if the triangle is obtuse, and on the longest side if the triangle is right.
6. If the triangle is equilateral, the circumcenter coincides with the centroid, orthocenter, and incenter of the triangle.
These properties make the circumcenter a crucial point of interest in studying and analyzing triangles.
What are the properties of the circumcenter of a triangle?
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1 answer