Note; Your teacher will grade your response to questions 13-15 to ensure you receive proper credit for your answer.
What are the properties of the circumcenter of a triangle?
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. It has the following properties:
1. The circumcenter is equidistant from all three vertices of the triangle. This means that the distances from the circumcenter to each of the vertices are equal.
2. The circumcenter is the center of the circumscribed circle, which is a circle that passes through all three vertices of the triangle. The radius of this circle is the same as the distance from the circumcenter to any of the vertices.
3. The circumcenter lies inside an acute triangle, on the midpoint of the hypotenuse of a right triangle, and outside an obtuse triangle.
4. If the triangle is equilateral, then the circumcenter coincides with the centroid, orthocenter, and incenter of the triangle.
5. The circumcenter is unique for any non-degenerate triangle, meaning that it is the only point that satisfies the conditions mentioned above.
These properties make the circumcenter an important concept in geometry, as it provides a way to define a unique circle that passes through all three vertices of a triangle.