Final answer:
The circumcenter of a triangle is the point where the perpendicular bisectors of the triangle's sides intersect. It is the center of the circumcircle, which passes through all three vertices of the triangle. The circumcenter can be inside, on, or outside the triangle depending on whether the triangle is acute, right, or obtuse.
Step-by-step explanation:
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. It can be described as option (c) – The point where the perpendicular bisectors of a triangle intersect. A perpendicular bisector of a side of a triangle is a line that is perpendicular to that side and divides it into two equal lengths. The circumcenter is equidistant from all three vertices of the triangle, which means that it is the center of the circumcircle, the circle that passes through all three vertices of the triangle.
This is different from the centroid or the center of mass of the triangle (b), which is the intersection of the medians, and also different from the orthocenter (d), which is the intersection of the altitudes of the triangle. Unlike the circumcenter, the centroid and the orthocenter have different properties and geometrical significance.
It’s important to note that the circumcenter’s location can vary based on the type of triangle. In an acute triangle, the circumcenter will lie inside the triangle; for a right triangle, it will be at the midpoint of the hypotenuse; and for an obtuse triangle, it will be outside the triangle. Regardless of its location, the circumcenter always has an equal distance to each of the triangle's vertices.