What are the points of concurrency in a triangle and what is mathematically special about each of them?

The **centroid** is the point of concurrency (intersection) of the three medians of a triangle. A median is a segment joining a vertex of a triangle with the midpoint of the opposite side.

*The centroid of triangle ABC is point D.*

An **orthocenter** is the point of concurrency of the three altitudes of a triangle. An altitude is the perpendicular line drawn from a vertex to the opposite side.

*The orthocenter of triangle ABC is point O.*

An **incenter** is the point of concurrency of the three angle bisectors of a triangle. An angle bisector is the ray which bisects an angle, or divides the angle into two congruent angles.

*The incenter of triangle ABC is point I.*

A **circumcenter** is the point of concurrency of the three perpendicular bisectors of a triangle. A perpendicular bisector is a segment which bisects a segment and forms right angles.

*The circumcenter of triangle ABC is point M.*

To see how these points of concurrency are constructed, please watch my short Screencast video.

*Now that you have seen how to construct a triangle’s points of concurrency using geometry, how could you use coordinate geometry and algebra to identify the points of concurrency?*

Given points A(10,7) B(10,1) and C(2,1), find the coordinates of the following:

- centroid
- orthocenter
- incenter
- circumcenter

Check your answers using this handy Point of Concurrency web calculator.

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## About Ms. Miles

I've taught mathematics for over 22 years. I love teaching math. Interests outside of teaching include learning and doing math, playing the piano, singing in a worship team, origami, kayaking, bicycling, swimming, collecting sea glass and shells.