Skip to main content
AMC 12 focus

Triangle Centers & the Euler Line

Centroid, incenter, circumcenter, and orthocenter — four points every triangle has, three of which are always collinear.

The four centers

  • Centroid (G) — intersection of the medians; the “center of mass.” In coordinates, it's just the average of the three vertices.
  • Incenter (I) — intersection of the angle bisectors; equidistant from all three sides, and the center of the inscribed circle.
  • Circumcenter (O) — intersection of the perpendicular bisectors of the sides; equidistant from all three vertices, and the center of the circumscribed circle.
  • Orthocenter (H) — intersection of the altitudes.

Coordinates and radii

G=(x1+x2+x33, y1+y2+y33)G = \left(\dfrac{x_1+x_2+x_3}{3},\ \dfrac{y_1+y_2+y_3}{3}\right)
r=Areas(inradius),R=abc4⋅Area(circumradius)r = \dfrac{\text{Area}}{s} \quad\text{(inradius)}, \qquad R = \dfrac{abc}{4\cdot\text{Area}} \quad\text{(circumradius)}

where ss is the semi-perimeter a+b+c2\tfrac{a+b+c}{2}.

The centroid has one more useful property: it divides each median in a fixed 2:12:1 ratio, measured from the vertex.

The Euler line

In any non-equilateral triangle, the centroid, circumcenter, and orthocenter are always collinear — this line is called the Euler line. Even more specifically, the centroid divides the segment from the orthocenter to the circumcenter in a fixed ratio:

HG=2⋅GOHG = 2 \cdot GO

(The incenter is generally not on the Euler line unless the triangle is isosceles or equilateral.)

Worked example

Problem: Triangle ABCABC has vertices A(0,0)A(0,0), B(6,0)B(6,0),C(0,8)C(0,8). Find the centroid.

Solution: Average the coordinates:

G=(0+6+03,0+0+83)=(2,83)G = \left(\dfrac{0+6+0}{3}, \dfrac{0+0+8}{3}\right) = \left(2, \dfrac83\right)

Practice problems

1.A triangle has vertices (1,2),(5,2),(1,8)(1,2), (5,2), (1,8). Find the centroid.

2.A triangle has sides 5, 12, 13 (a right triangle). Find the circumradius.

3.A triangle has sides 6, 8, 10 and area 24. Find the inradius.

4.In a triangle, GO=5GO = 5 where GG is the centroid and OO the circumcenter. Find HOHO (distance from orthocenter to circumcenter), given H,G,OH, G, O lie on the Euler line in that order.

Special Segments in TrianglesArea Techniques