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# Centers of a Triangle

## Incenter

Let's construct a triangle and bisect each angle.

Notice that all three angle bisectors intersect in the same point. This
is the **incenter** and is labeled **I**. When any two lines intersect
they are said to be concurrent. Any two lines not parallel will have a point
of concurrency, but it is special for three lines to meet in the same point.
In this triangle, the point is the same distance from each side; so it is
the center of a circle that can be inscribed in the triangle.A radius is
formed by constructing a perpendicular to one side. The circle touches all
three sides, but does not extend outside them.

Investigate: What happens if the triangle is an obtuse one? What if it
is a right triangle?

Make a conjecture about the incenters of various shapes of triangles.

### Orthocenter

Constuct another triangle and the altitudes which are perpendicular lines
from the vertex to the line of the opposite side.

Notice again that the three lines intersect in the same point. This is
called the **orthocenter** and is usually labeled **H**. Try this
with an obtuse

Investigate: What happens if the triangle is obtuse? Hint: the sides
of the triangle may need to be extended. What if it is a right triangle?

Make a conjecture about the orthocenters of the various shapes of triangles.

### Centroid

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Draw another triangle and construct the three medians, the segments from
each vertex to the middle of the opposite side.

Again the three constructed lines intersect or concurr; the point of
intersection is called the **centroid** and is labeled **G**. It divides
each median into two parts so that the distance from the centroid to the
vertex is two-thirds the distance from the centroid to the midpoint.

Investigate: What happens in obtuse triangles? What about right triangles?
Make a conjecture about the centroids of triangles.

### Circumcenter

There is one more type of triangle center to explore. Construct the perpendicular
bisectors of each side of a triangle.

Once more the constucted lines intersect at a point; it is called the
**circumcenter** and is labeled **C.** The circumcenter is the center
of a circle which is circumscribed about the triangle. It touches each vertex
of the triangle; a radius is a segment from the circumcenter to a vertex.
Notice below.

Investigate the circumcenter of obtuse triangles and right triangles.
Make a conjecture about the circumcenters of various shapes of triangles.

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