Centers of a Triangle

by Beverly Hales

Centroid

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.

GSP sketch

Orthocenter

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

The orthocenter is not always in the interior of the triangle. Choose a vertex of the triangle and translate the point to view the various location of the orthocenter.

GSP sketch

Circumcenter

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.

GSP sketch

Circumcircle

The Circumcircle is the circumsribed circle of a given triangle.

The circumcenter is the the center of the circle.

GSP sketch

Incenter

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 bisector lies on the angle bisector, then the incenter must be on the angle bisector of each angle of the triangle.

GSP sketch

Euler Line

A segment that contains the circumcenter, centroid, and orthocenter.

GSP sketch

Medial Triangle

A triangle constructed by connecting the three midpoints of a given triangle.

The Euler line is the same for both of the triangles.

GSP sketch

Nine Point Circle

The nine point circle exist for any triangle. The nine-point circle is constructed using these points associated with the triangle:

*the midpoints of the sides

*the feet of the altitudes

*the midpoints of the segments from the orthocenter to the vertices.

GSP sketch

Return to Home page

Return to Final Project