Triangle Centers

Centroid, circumcenter, incenter, orthocenter — four triples of lines, four meeting points.

The idea

Theorem (The four centers of a triangle).

In every triangle $ABC$, each of the following triples of lines meets at a single point: the three medians, each joining a vertex to the midpoint of the opposite side, meet at the centroid; the three perpendicular bisectors of the sides meet at the circumcenter; the three angle bisectors meet at the incenter; and the three altitudes, each dropped from a vertex perpendicular to the line of the opposite side, meet at the orthocenter.

Three arbitrary lines need not share a point; that each of these triples does is the content of the theorem.

The centroid $G$ divides every median in the ratio $2:1$ measured from the vertex, and its coordinates are the average of the vertices' coordinates — it is the triangle's balance point.

A point on a side's perpendicular bisector is equidistant from that side's endpoints, so the circumcenter is equidistant from all three vertices: it is the center of the circle through them.

A point on an angle's bisector is equidistant from the angle's two sides, so the incenter is equidistant from all three sides: it is the center of the inscribed circle.

The four coincide only in an equilateral triangle; in general they are four distinct points.

Ways to work on it

• Walkthrough. Centroid from coordinates, the 2:1 ratio, and the four centers.
• Proof. See why the three medians meet at a single point.
• Practice. Centroids from coordinates and which-center-does-what.
• Hardest. Invert the 2:1 ratio, and find a right triangle's circumcenter.

Not sure where to start? Take the ten-question placement test.