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Point of Concurrency - Concept

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Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

A point of concurrency is where three or more lines intersect in one place. Incredibly, the three angle bisectors, medians, perpendicular bisectors, and altitudes are concurrent in every triangle. There are four types important to the study of triangles: for angle bisectors, the incenter; for perpendicular bisectors, the orthocenter; for the altitudes, the circumcenter; for medians, the centroid.

A point of concurrency is a place where three or more, but at least three lines, rays, segments or planes intersect in one spot. If they do, then those lines are considered concurrent, or the the rays are considered concurrent. So let's look at two examples here.
If I look at this example right here we have three lines and in this spot right here we have two lines intersecting, in this spot we have two lines intersecting and here we have two lines intersecting. So none of these lines are concurrent. There is no point of concurrency.
If we look at these three lines right here, it's pretty clear to see that they all intersect in one point. So that point right there where three lines intersect would be our point of concurrency.
But why does this matter? Well, it matters in triangles when we're talking about four types of points of concurrency. The first one is formed by the three angle bisectors.
So if you're thinking about a triangle, if you're to construct a three angle bisectors, you would be constructing a special point of concurrency known as the in center, and the in center is the center of a circle that when you draw that circle it will intersect the sides exactly one time.
If you were to construct the three perpendicular bisectors of each of the three sides, then you will be finding the point of concurrency called the "circumcentre." And a circumcentre is like the in centre except for the circumcentrer circle intersects the three vertices not the three sides.
The next type is the three altitudes. So if you took your triangle and constructed your three altitudes, you'd be constructing a point of concurrency known as the "orthocenter."
The last type is a median. So if you constructed this three medians of each side connecting the vertex to the midpoint, then you'd be constructing the centroid, which is also the center of gravity or the center of mass for a given triangle.
So the reason why points of concurrency is an important vocab word is because there are four major types of points of concurrency or talking about triangles.

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