Orthocenter, Centroid, Circumcenter and Incenter of a Triangle
The orthocenter is the point of intersection of the three heights of a triangle.
A height is each of the perpendicular lines drawn from one vertex to the opposite side (or its extension).
The centroid is the point of intersection of the three medians.
A median is each of the straight lines that joins the midpoint of a side with the opposite vertex
The centroid divides each median into two segments, the segment joining the centroid to the vertex multiplied by two is equal to the length of the line segment joining the midpoint to the opposite side.
BG = 2GA
The circumcenter is the point of intersection of the three perpendicular bisectors.
A perpendicular bisectors of a triangle is each line drawn perpendicularly from its midpoint.
The circumcenter is the center of a triangle's circumcircle (circumscribed circle).
The incenter is the point of intersection of the three angle bisectors.
The angle bisectors of a triangle are each one of the lines that divide an angle into two equal angles.
The incenter is the center of the circle inscribed in the triangle.
Line of Euler
The orthocenter, the centroid and the circumcenter of a non-equilateral triangle are aligned; that is to say, they belong to the same straight line, called line of Euler.