Understanding the Points of Concurrency in a Triangle

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 21 years. I love teaching math. Interests outside of teaching include learning and doing math, playing the piano, singing in a praise band, origami, kayaking, bicycling, swimming, collecting sea glass and shells.
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