"Centers" of triangles

Various different kinds of "centers" of a triangle can be found.  This page summarizes some of them.

Click any of the diagrams below, to go to the page that talks about that triangle center.

For any triangle all three medians intersect at one point, known as the centroid.

The three altitudes (lines perpendicular to one side that pass through the remaining vertex) of the triangle intersect at one point, known as the orthocenter of the triangle.

The three perpendicular bisectors of the sides of the triangle intersect at one point, known as the circumcenter - the center of the circle containing the vertices of the triangle.

The three angle bisectors of the angles of the triangle also intersect at one point - the incenter, and this point is the center of the inscribed circle inside the triangle.

I believe all of these can be proved using vectors and also expressions for finding these points in any triangle can be found.

For the centroid in particular, it divides each of the medians in a 2:1 ratio. Sorry I don't know how to do diagrams on this site, but what I mean by that is: let the triangle be ABC, the median though A be AM (M lying on BC), and the centroid be point D. Then AD:DM=2:1 and this is true for the other two medians as well.

The orthocenter and circumcenter are isogonal conjugates of one another.

Internet References

Related pages in this website:

Barycentric Coordinates, which provide a way of calculating these triangle centers (see each of the triangle center pages for the barycentric coordinates of that center).

Summary of geometrical theorems summarizes the proofs of concurrency of the lines that determine these centers, as well as many other proofs in geometry.

The webmaster and author of the Math Help site is Graeme McRae.
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