The Four Triangle Centers
Locate and interpret the circumcenter (equidistant from vertices), incenter (equidistant from sides), centroid (balancing point, 2:1 ratio on each median), and orthocenter. Apply the centers to solve problems.
A lesson of Geometry, in Triangle Relationships, about 10 minutes.
By the end of this lesson, you will be able to locate and describe the four classical triangle centers — centroid, circumcenter, orthocenter, and incenter — identify their key properties, and use coordinates to find the centroid and circumcenter.
The Big Idea
The four classical triangle centers each arise from a different set of special segments. Each has a unique property — one balances the triangle, one circumscribes it, one inscribes a circle within it. Their positions relative to the triangle depend on the triangle's shape, and their relationships to each other reveal the deep structure of triangles.
The Centroid
Formed by: the three medians
Location: always inside the triangle
Key property: the centroid is the triangle's center of mass — if you could cut a triangle from cardboard, it would balance on the centroid.
Centroid ratio: each median is divided 2:1 from vertex to midpoint.
Finding with coordinates: average the three vertices.
$$G = \left(\frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}\right)$$
The Circumcenter
Formed by: the three perpendicular bisectors of the sides
Location: inside for acute triangles, on the hypotenuse for right triangles, outside for obtuse triangles
Key property: equidistant from all three vertices → center of the circumscribed circle (circumcircle). The circumradius $R$ satisfies $PA = PB = PC = R$.
Finding with coordinates: solve two perpendicular bisector equations simultaneously.
The Orthocenter
Formed by: the three altitudes
Location: inside for acute triangles, at the right angle vertex for right triangles, outside for obtuse triangles
Key property: no simple "center of" interpretation, but central to advanced geometry. The orthocenter, centroid, and circumcenter always lie on a line called the Euler line.
The Incenter
Formed by: the three angle bisectors
Location: always inside the triangle
Key property: equidistant from all three sides → center of the inscribed circle (incircle). The inradius $r$ is the perpendicular distance from the incenter to each side.
$$r = \frac{\text{Area}}{s} \quad \text{where } s = \frac{a+b+c}{2} \text{ (semi-perimeter)}$$
The Euler Line
The centroid $G$, circumcenter $O$, and orthocenter $H$ are collinear. Euler proved that: