the center of the circumcircle of a triangle is called the circumcenter of the triangle.

the center of the circumscribed circle of a triangle is the intersection point of the perpendicular bisectors of the three sides of the triangle. the three vertices of the triangle are on this circumscribed circle.

the intersection point of the perpendicular bisectors of the three sides of a triangle is called the circumcenter of the triangle.

the distances from the circumcenter to the three vertices are equal.

the circle passing through each vertex of the triangle is called the circumcircle of the triangle. the center of the circumscribed circle is the circumcenter of the triangle. this triangle is called the inscribed triangle of the circle.

(1) the circumcenter of an acute-angled triangle is within the triangle;

(2) the circumcenter of the right triangle is on the hypotenuse and coincides with the midpoint of the hypotenuse;

(3) the circumcenter of an obtuse triangle is outside the triangle.

(4) the circumcenter and incenter of an equilateral triangle are the same point.