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 with three medians drawn, intersecting at a point.

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

images/triangle.js?mode=median

Try this: cut a triangle from cardboard, draw the medians. Do they all meet at one point? Can you balance the triangle at that point?

Triangle with three medians intersecting at the centroid.

The centroid divides each median in the ratio 2 : 1

centroid dividing a median in the ratio 2 to 1

Try it yourself on any of the three medians.

Centroid Ratio Calculator

If the Total Median length is:

Vertex to Centroid (2)

Centroid to Side (1)

Example: How high up this triangle is the Centroid?

18 cm is the height (altitude)

Simply one third of the way up!

So 18 cm3 = 6 cm up from base

Circumcenter

Triangle with three perpendicular bisectors drawn, intersecting at a point.

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":

images/triangle.js?mode=circum

Try this: drag the points above until you get a right triangle (just by eye is OK). Where is the circumcenter? Why?

Incenter

Triangle with three angle bisectors drawn, intersecting at a point.

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":

images/triangle.js?mode=incircle

Try this: work out the incenter of a triangle using a compass and straightedge at: Inscribe a Circle in a Triangle

Orthocenter

Triangle with three altitudes drawn, intersecting at a point.

Draw a line segment (called the "altitude") at right angles to a side that goes to the opposite corner.
Where all three lines intersect is the "orthocenter":

images/triangle.js?mode=ortho

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.