Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

## Point of Concurrency (Definitions, Bisectors, & Examples)

A key part of learning is adding to your vocabulary. Concurrency is an excellent word to learn in geometry. In mathematics, it means a point shared by three or more lines. This lesson focuses on points of concurrency in triangles.

## What you'll learn:

After working your way through this lesson and video, you will be able to:

• Define the geometry term "concurrency"
• Recall and define the four different kinds of points of concurrency for triangles, which are the centroid, circumcenter, incenter and the orthocenter
• Identify the four points of concurrency in a drawing of a triangle
• Recall and state that two of the four points of concurrency, the circumcenter and orthocenter, may fall outside the triangle

## Make It!

Take three, uncooked, spaghetti strands or three pick-up sticks, hold them in one hand, and drop them onto a flat, hard surface. You may need to try a few times, but when you get the three sticks to all cross each other, the point where they cross is a point of concurrency.

## Point of Concurrency Definition

A point of concurrency is a single point shared by three or more lines. Constructed lines in the interior of triangles are a great place to find points of concurrency.

When you construct things like medians, perpendicular bisectors, angle bisectors, or altitudes in a triangle, you create a point of concurrency for each one.

Since you can construct four different types of line segments for the triangle, you can have four different points of concurrency. Each point of concurrency is associated with the intersection of a particular type of line segment:

1. Centroid -- medians
2. Circumcenter -- perpendicular bisectors
3. Incenter -- angle bisectors
4. Orthocenter -- altitudes

Beware! Even though you create the line segments (or lines) using parts of the triangle, two of the four points of concurrency do not have to be in the triangle's interior! The circumcenter and orthocenter can lie inside or outside the triangle.

## What is a Triangle?

A triangle is a two-dimensional polygon with three straight sides closing in a space. Its interior has three interior angles and a measurable area. You can bisect (cut in half) the interior angles and the sides. An angle bisector divides an angle; a median and perpendicular bisector divides a side.

## Centroid of a Triangle

The median of a triangle's side is a line segment drawn from the side's midpoint to the opposite angle. A median bisects a side. A triangle has three medians. The median creates two smaller triangles of equal area inside the original triangle. Another way to say that is the median divides the triangle's area in half.

The point of concurrency shared by the three medians is the triangle's centroid. If the triangle is an actual, physically existing triangle, the centroid is also the triangle's center of mass, or center of gravity. You could balance the triangle on a pencil point at the centroid!

The centroid is always within the triangle. The centroid is the point exactly two-thirds of the distance along each median. This sets up the ratio $2:1$, with the longer portion of each median closer to each vertex.

## Incenter of a Triangle

The interior angles of triangles can be bisected with an angle bisector, a line segment originating at the vertex and extending to the opposite side. The point where the three angle bisectors of a triangle cross one another is the triangle's incenter. It is also the center point of the triangle's incircle.

An incircle is the largest circle that can be drawn inside the triangle while touching all three sides. The sides will be tangent to the circle. An incenter always lies within the triangle.

## Circumcenter of a Triangle

Using the midpoint of each side does not obligate you to stretch a line segment to the opposite angle. What if you, instead, constructed a perpendicular line or line segment from that midpoint? You would create a perpendicular bisector.

Construct three perpendicular bisectors and they will cross each other at the point of concurrency called the circumcenter.

A circumcenter is the center of a circle passing through all three vertices of the triangle. Such a circle is called a circumcircle.

Because the line segments must be perpendicular to the midpoint of each side, you construct them without regard for the vertices. For this reason, the circumcenter may lie inside or outside the triangle. The circumcenter and orthocenter are the two points of concurrency that can do that.

## Orthocenter of a Triangle

A perpendicular line from a triangle's side to the opposite vertex gives you the triangle's height, or altitude. Where the three altitudes of a triangle meet, that point of concurrency is called the orthocenter.

"Ortho" is a Greek prefix that means "upright," "correct," or "right." You visit the orthopedist to straighten out the bones in your feet. You visit the orthodontist to straighten out your teeth. An orthocenter is the single common point along three orthogonal (upright) lines.

Because an altitude might lie outside the triangle, the orthocenter might also fall outside the triangle. Here are two triangles, $△UPS$ and $△DWN$:

[insert triangles as described below]

The first triangle, $△UPS$, is an predictable, ordinary, acute triangle with predictable, ordinary altitudes all safely drawn in the interior. The predictable, ordinary orthocenter is inside the triangle.

But the second triangle -- whoa! -- it is a very, very obtuse triangle! Two of its three altitudes are outside the triangle, because two of its angles are very acute. The orthocenter is also outside the triangle!

## Unusual Facts

Mathematicians have discovered interesting facts about the points of concurrency that are always true:

1. For any right triangle, the orthocenter is always at the vertex of the right angle.
2. The Euler line -- named for Leonhard Euler, 18th century Swiss mathematician -- is a line that always contains the orthocenter, centroid, and circumcenter for any triangle. They are collinear (lying along one line).
3. A triangle's centroid will always fall between the triangle's orthocenter and circumcenter.
4. The distance from centroid to circumcenter is always half the distance between the centroid and orthocenter.
5. For every acute triangle, the circumcenter is always inside the triangle.
6. For every obtuse triangle, the circumcenter is always outside the triangle and always opposite the obtuse angle. The orthocenter is also always outside the triangle, opposite the longest leg.
7. For every right triangle, the circumcenter is always the midpoint of the hypotenuse.
8. All four points of concurrency can themselves be concurrent! Only with an equilateral triangle will the centroid, circumcenter, incenter and orthocenter always be the same point!

## Lesson Summary

Now that you have made your way through the reading, studied the drawings, and watched the video, you are able to define the geometry term "concurrency," recall and define the four different kinds of points of concurrency for triangles (centroid, circumcenter, incenter, and orthocenter), and identify the four points of concurrency in a drawing of a triangle. You also know that two of the four points of concurrency -- the circumcenter and the orthocenter -- may be found outside the triangle.

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