Investigation of Centers of a Triangle

by

Molade Osibodu

** Objective:** To show the construction of various centers of a triangle.

**CENTROID (G)**

The centroid of a triangle is the intersection of the three medians of the triangle. The median is the segment from the vertex to the midpoint of the opposite side.

To construct, first create the midpoint on each side of the triangle, then construct a line from the midpoint to the opposite side as shown in figure 1 below:

Figure 1: The center G, is the centroid of the triangle. Click here for GSP

**ORTHOCENTER (H)**

The orthocenter of a triangle is the intersection of the three lines containing the altitudes. The altitude is the perpendicular segment from the vertex to the line of the opposite side.

To construct, create the perpendicular line from the vertex to the opposite side as shown in figure 2 below:

Figure 2: The center H is the orthocenter of the triangle. Click here for GSP

**CIRCUMCENTER (C)**

The circumcenter of a triangle is the point in the plane equidistant from the three vertices of the triangle.

To construct, create the midpoint of the sides and construct the perpendicular line from the midpoint and the side on which the midpoint lies as shown in figure 3 below:

Figure 3: The center C is the circumcenter of the triangle. Click here for GSP

**INCENTER (I)**

The incenter of a triangle is the point on the interior that is equidistant from the three sides.

To construct, create the angle bisector by clicking on each of the vertices of the triangle in different orders as shown in figure 4 below:

Figure 4: The center I is the incenter of the triangle. Click here for GSP

**NINE - POINT CIRCLE (N)**

The nine-point circle for any passes the three midpoints of the sides, the feet of the altitudes and the three midpoints of the segments from the respective vertices to the orthocenter.