Geometry

Question Analysis

This question is based on the Triangle Perpendicular Bisector Concurrence Theorem. The key is to identify the type of center point D represents for the triangle inscribed in the circle, by understanding the properties of different triangle centers.

 

Key Concept Explanation

1. Circumcenter: It is the point where the perpendicular bisectors of the sides of a triangle intersect. It is the center of the circum - circle, which is the circle that passes through all three vertices of the triangle.

2. Incenter: The incenter is the point of intersection of the angle bisectors of a triangle. It is the center of the incircle, which is tangent to all three sides of the triangle.

3. 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.

4. Orthocenter: The orthocenter is the point of intersection of the altitudes of a triangle. An altitude is a perpendicular line from a vertex to the opposite side.

 

Step-by-Step Solution

1. Observe that point D is the center of the circle that passes through all three vertices of the triangle.

2. Recall the definitions of different triangle centers. Since the circumcenter is the center of the circle that circumscribes the triangle (passes through all vertices), point D must be the circumcenter.

3. Eliminate the other options:

The incenter is related to the incircle (tangent to sides), not the circum - circle, so it's not the incenter.

The centroid is related to medians and has no direct connection to the circle passing through vertices, so it's not the centroid.

The orthocenter is related to altitudes and not the circum - circle, so it's not the orthocenter.

 

Click "Show Answer" to reveal the answer and analysis