**Jackie Gammaro**

**Centers of Triangles**

We have already studied the **centroid** and how it is a point of balance for a triangle. We then took a detailed look at
the location of **orthocenters**.

Lets look again at these centers and also the **Circumcenter** , the **circumcircle,** the **incenter**,
the i**ncircle,** and finally the

**I – Centroid**

**Vocab: Median –
**the median of a triangle is the segment from a vertex to the midpoint of
the opposite side

**How to Construct a
Centroid:**

1. Construct triangle ABC

2. Construct
the midpoint of each side of the triangle. Let the midpoint of **AB**
is D, the midpoint of** BC** is E, and
the midpoint of **AC** is F.

3. Connect
midpoint D with vertex C, to create median **DC
**and likewise, create medians **EA**
and **FB**.

4. The
**centroid** is the point of concurrency
for the three medians, point **G**.

**II) Orthocenter**

**III) Circumcenter and
Circumcircle**

**Vocab: circumcenter – **the point in
the plane equidistant from the three vertices of the triangle.

The circumcenter sits on the three perpendicular bisectors of a triangle.

The circumcenter is also the center of the **circumcircle** – the circumscribed
circle of the triangle.

**Constructing a
Circumcenter**

1. Construct triangle ABC.

2. Construct perpendicular bisectors of each side of the triangle, by constructing the midpoint of each side, then constructing perpendicular lines to each side through each midpoint.

3. The
point of concurrency for the three perpendicular bisectors is the **circumcenter**.

4. To
construct a **circumcircle,** you know
the vertices of the triangle lie on the circumference of the
circumcircle.- construct a circle
that has center at the circumcenter and a point on the circle pass through one of the triangles vertices.

**Creating an Incenter:**

The *Incenter* is
the point of concurrency for the three angle bisectors of each angle of the
triangle.

To create the **incenter**,
you have to create the three angle bisectors for each triangle. If I recall back to high school
geometry, I remember how to create an angle bisector, by creating an arc that
passes through both side of the angle.

Then create another arc, along one side of the angle, create an equidistant arc on the other side of the angle. Where those two arcs intersect is the a point that lies on the angle bisector of the given angle.

Shown here is triangle ABC, with incenter, I. The circle incscribed within the triangle is called the incircle. Recall, that the incenter is the point on the interior of the triangle that is equidistant from the three sides. That being said, then the incircle's radius has to be perpendicular to each side at the circle's and triangle's point of tangency. The dashed lines are perpendicular lines created, the points of intersection with the perpendcular lines and the sides of the triangle are points on the incircle's circumference. The orange circle is the incircle.

__Euler Line__

Here is shown the centers of triangle DEF.

H - orthocenter

I - Incenter

G - Centroid

C - Circumcenter

Notice H, G and C are collinear. The line they sit along is
called the** Euler Line.**