Triangle Centers

Where is the center of a triangle?

There are actually thousands of centers!

Here are the 4 most popular ones:

triangle centers: Centroid, Circumcenter, Incenter and Orthocenter

Centroid, Circumcenter, Incenter and Orthocenter

For each of those, the "center" is where special lines cross, so it all depends on those lines!

Let's look at each one:

Centroid

triangle center median

Draw a line (called a "median") from a corner to the midpoint of the opposite side.
Where all three lines intersect is the centroid, which is also the "center of mass":

Circumcenter

triangle center circumcenter Draw a line (called a "perpendicular bisector") at right angles to the midpoint of each side.
Where all three lines intersect is the center of a triangle's "circumcircle", called the "circumcenter":

Incenter

triangle center angle bisector Draw a line (called the "angle bisector") from a corner so that it splits the angle in half
Where all three lines intersect is the center of a triangle's "incircle", called the "incenter":

Orthocenter

triangle center height Draw a line (called the "altitude") at right angles to a side and going through the opposite corner.
Where all three lines intersect is the "orthocenter":

Note that sometimes the edges of the triangle have to be extended outside the triangle to draw the altitudes. Then the orthocenter is also outside the triangle.