Ideas in Geometry/Instructive examples/Section 3.1

#23 The Euler Line

The Euler line is the line connecting three certain points within a triangle. These points are the circumcenter, the centroid and the orthocenter. The circumcenter is the point where the perpendicular bisectors of the sides of a triangle meet. The centroid is the point where the lines of a triangle that connect the vertexes to the midpoints of the opposite sides of a triangle meet. The orthocenter is the point where the altitudes of a triangle meet. The altitude is the line segment originating at a vertex of the triangle that meets the line containing the opposite side at a right angle. When all of these points are connected, they form a line which is the Euler line. The centroid lies a third of the distance from the circumcenter to the orthocenter. See figures below.