The Centers of a Triangle

By: Diana Brown

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

Click on the picture to open the GSP file to explore what happens to the figure as you drag one of the vertices.

Notice that the centroid is always on the inside of the circle.

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

Click on the first picture to open the GSP file to explore what happens to the figure as you drag one of the vertices.

Notice that the orthocenter is sometimes outside the triangle, sometimes on the triangle, and sometimes inside the triangle.

The CIRCUMCENTER of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector of the segment determined by the two points, the Circumcenter is on the perpendicular bisector of each side of the triangle.

Click on the first picture to open the GSP file to explore what happens to the figure as you drag one of the vertices.

Notice that the Circumcenter is also sometimes outside the triangle and sometimes inside the triangle.

The INCENTER of a triangle is the point on the interior of the triangle that is equidistant from the three sides. Since a point interior to an angle that is equidistant from the two sides of the angle lies on the angle bisector, then the Incenter must be on the angle bisector of each angle of the triangle.

Click on the picture to open the GSP file to explore what happens to the figure as you drag one of the vertices.

Notice that the Incenter stays inside the triangle.

Summary Table:

 Concurrent Inside/Outside/On the Triangle Center Lines Acute Right Obtuse Centroid Medians inside inside inside Orthocenter Altitudes inside on outside Circumcenter Perpendicular bisectors of the sides inside on outside Incenter Angle Bisectors inside inside inside