**Assignment
#4**

__Centers
of Triangles__

By

**Michelle
Nichols**

__Centroid__

The Centroid of a triangle is the intersection
of the medians, or the center of gravity.
It is formed by finding the midpoints of the triangle, connecting each
midpoint to the vertices, and connecting their intersections.

__Incenter__

The 3 angle bisectors of a triangle intersect at a point called
the Incenter of a triangle. It is the center of the largest circle that
will fit inside the triangle. Look at
the Incenter:

__Circumcenter__

The circumcenter is the intersection of
all 3 perpendicular bisectors of the sides of a triangle. It’s called the circumcenter
because it is the center of a circle that passes through all of the vertices of
the triangle.

__Orthocenter__

The Orthocenter is the intersection of the 3 altitudes of a
triangle. An altitude of a triangle is a
segment that is drawn so that is passes through the vertex and is perpendicular
to the opposite side.

__Euler Line__

The centroid, circumcenter,
and orthocenter of a triangle are all collinear. The Euler Line is the line that connects all
of these points.

__Exploration__

Take any triangle. Construct
a triangle connecting the 3 midpoints of the sides. This is called the MEDIAL triangle. It is similar to the original triangle and ¼ of
its area. Construct G, H, C, and I for
this new triangle. Compare to G, H, C,
and I in the original triangle.

Now if we construct the Euler Line for both, thus observing G, H,
and C, let’s see how they compare.

Large Triangle Medial Triangle

The points lie on the same line, however, the locations and distances
between each change. The larger
triangles’ Euler Line appears to be twice the distance. When measurements and ratios are taken, we
find this to be exactly true. The Larger
triangle Euler Line has a 2:1 ratio to the medial triangles’ Euler line. Also, the centroid
and orthocenter swap sides.

Observe the
triangles together: