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Properties of Concurrent Lines in a Triangle

Properties of Concurrent Lines in a Triangle
Coming up next: Applications of Similar Triangles

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  • 0:06 Where Lines Meet
  • 0:55 Centroid
  • 1:34 Orthocenter
  • 3:06 Incenter
  • 4:48 Circumcenter
  • 5:33 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Centroids, orthocenters, incenters, circumcenters, oh my! Don't worry though. In this lesson, we master the various terms for concurrent lines in triangles and match them to altitudes, angle bisectors, perpendicular bisectors and medians.

Where Lines Meet

If you've ever been to New England, you may have encountered their crazy intersections. Instead of two roads meeting, which is normal and functional, they might have three or four roads meet, often at weird angles. This is can seem chaotic and strange.

The worst part is they're not big into signage. So you may not even know what to call the roads. But if those roads were lines in a triangle, they'd definitely have names, like median and angle bisector.

Even better, the points where those roads meet would have names, too. That's what we're going to learn here. We call these intersections points of concurrency. Concurrence is when three or more lines meet at a single point. We can call lines like these concurrent lines. Let's look at a few.

Centroid

First up, let's look at medians. These are lines drawn from the vertices of a triangle that bisect the opposite sides. Medians are so neat and orderly, splitting those opposite sides perfectly in half. They're like the medians in a road with their lovely shrubbery.

The points where three median lines are concurrent, or intersect, is called a centroid. Centroid also means the center of mass. And, in fact, if you took your triangle and tried to balance it on a pencil, the centroid is where you should put your pencil point. That will perfectly balance the mass of the triangle.

Orthocenter

And then there are altitudes. An altitude is a perpendicular line segment drawn from a vertex to the opposite side. It's like the altitude of a mountain - that's the distance from the summit straight down, or perpendicular, to sea level.

The point where three altitude lines are concurrent is called the triangle's orthocenter.

Orthocenter? That's a weird word. It sounds like orthodontist. Oh, man, I don't like to be reminded of my orthodontist. He tried to make my teeth straight by inflicting pain and marring my smile with braces throughout high school. But orthocenter and orthodontist share something. The root ortho- means straight or right. Just like an orthodontist straightening your teeth, so they're at right angles in your mouth, an orthocenter is the center of right-angled lines in a triangle.

Granted, if this triangle with altitudes drawn in it and an orthocenter here was your mouth, you'd definitely need to see an orthodontist.

One fun thing about orthocenters is that they don't need to be inside a triangle. Consider this triangle. If we draw an altitude from C, it hits AB. That's normal. What about from A? That's going to be perpendicular to BC out here. And from B? That's perpendicular to AC down here. So we need to extend our altitude line from C down to meet the other two lines, and our orthocenter is all the way out here.

Some orthocenters can be found on the outside of a triangle
External orthocenter example for a triangle

Incenter

Next, let's look at angle bisectors. These are lines drawn from an angle that bisect the angle, or splits it in half. That's easy to remember. What they do is right there in their name - they bisect the angle, so we call them angle bisectors.

That's like North St. actually going north. I think that happens sometimes in New England. Though East Boston is actually north of the North End, which is super confusing.

But angle bisectors - they always meet inside a triangle. And watch this. Let's draw an inscribed circle in our triangle. That's a circle that touches, or is tangent to, all three sides of the triangle. The point of concurrence for our angle bisectors is also the center of the inscribed circle. Therefore, we call the point where three angle bisectors are concurrent the incenter. The incenter is in the center of the inscribed circle.

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