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 has the following properties:
1. It is the point of intersection of the perpendicular bisectors of the sides of the triangle. This means that the distance from the circumcenter to each of the triangle's vertices is equal.
2. It is equidistant from the three vertices of the triangle. This means that the circumradius, which is the distance from the circumcenter to any of the triangle's vertices, is constant.
3. The circumcenter lies inside an acute triangle, on the centroid of an equilateral triangle, and outside an obtuse triangle.
4. In an equilateral triangle, the circumcenter coincides with the centroid, incenter, orthocenter, and the intersection point of the medians, altitudes, and angle bisectors.
5. The circumcenter of a right-angled triangle lies on the midpoint of the hypotenuse.
6. The circumcenter may lie outside the triangle if the triangle is obtuse.
7. It plays a key role in the construction of the circumcircle, which is the circle passing through all three vertices of the triangle. The circumcircle has the circumcenter as its center and the circumradius as its radius.