The centroid is the point on a triangle where the three concurrent median lines of a triangle are intersect. The median is the line segment that joins the vertex of a triangle to the opposites side; the vertex is the point at which a new side of the triangle beings from a previous one. In a triangle where the sides are denoted using A, B and C, the centroid is normally noted using the letter G.
The circumcenter is the point on a triangle where perpendicular bisectors of the three sides are concurrent. The perpendicular bisectors of a triangle are the lines that divide the individual sides into two equal (also called congruent) parts. These lines are called perpendicular because they intersect with one another at a 90-degree angle. The circumcenter is normally noted using the letter O.
The orthocenter is the point on a triangle in which the three lines of altitude are concurrent. Lines of altitude are the lines on each side of the triangle that run from the highest point. Much like the other points of concurrency, there is always an orthocenter, no matter where the vertices of the triangle are or how long the sides are. The orthocenter is normally denoted with the letter H.
Any two lines that are not parallel are concurrent. Finding the point of concurrency for two lines is relatively easy. All you need to do is take the equations for the two lines and set up a system of equations. First solve for "x" to find the point at which the lines cross on the x-axis, and then solve for "y" to find the point at which the lines cross on the y-axis. This coordinate is the concurrent point; if the equation ends up with an unsolvable answer for x or y, then the lines are parallel.