Final answer:
The circumcenter is the center of a triangle's circumcircle and can be inside, on the edge, or outside the triangle. The circumcenter is found at the intersection of the sides' perpendicular bisectors. The incenter, often misstated as 'intercenter', is where the angle bisectors intersect and is the center of the incircle.
Step-by-step explanation:
The question involves the concepts of geometry specific to triangles and their centers. Let's address each point:
- (a) The circumcenter is indeed the center of a triangle's circumcircle, the unique circle in which all the vertices of the triangle lie on the circle's circumference.
- (b) The statement that the circumcenter is always located inside the triangle is incorrect. The circumcenter can be inside, on, or outside the triangle, depending on whether the triangle is acute, right, or obtuse, respectively.
- (c) The perpendicular bisectors of the sides of a triangle do indeed meet at the circumcenter. This is one of the concurrent lines in a triangle.
- (d) There seems to be a typo; 'intercenter' should be 'incenter,' which is the center of a triangle's incircle, and it is the point where the angle bisectors of the triangle intersect. The incircle is the largest circle that fits entirely inside the triangle.
The foci of a circle are not mentioned in the context of triangle centers, but for clarification, the foci of a circle are at the same point, which is the center of the circle.