Geometry

Question Analysis

This question revolves around the Triangle Perpendicular Bisector Concurrence Theorem. The key is to identify the point that has equal distances to all vertices of a triangle among the four given triangle centers.

Key Concept Explanation

1. Centroid: The centroid is the point of intersection of the medians of a triangle. A median connects a vertex to the mid - point of the opposite side.

2. Incenter: The incenter is the point of intersection of the angle bisectors of a triangle. It is equidistant from the sides of the triangle.

3. Circumcenter: The circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle. By the property of the perpendicular bisector, any point on the perpendicular bisector of a line segment is equidistant from the two endpoints of that segment. Since the circumcenter lies on the perpendicular bisectors of all sides of the triangle, it is equidistant from the vertices.

4. Orthocenter: The orthocenter is the point of intersection of the altitudes of a triangle.

Step-by-Step Solution

A. Centroid: Incorrect. It divides each median in a 2:1 ratio and is not equidistant from vertices.

B. Incenter: Incorrect. It is the center of the incircle and is equidistant from the sides of the triangle.