Centers of Circles

Kasey Nored

This write-up is an exploration
targeted for high school students.

Centroid

A centroid
of a triangle is the common intersection of the three medians of the triangle. The median is constructed by finding the midpoint of a side of a
triangle and connecting the opposite vertex to the midpoint. A centroid will remain inside the circle for the three types
of triangles categorized by their angles.

Orthocenter

Orthocenters are found by the intersection of the
altitudes of a triangle. Altitudes
are found by creating a line perpendicular to the line containing the side of
the triangle that crosses through the opposite vertex. You will notice that orthocenters, unlike centriods,
can be inside, on or outside the triangle.

Circumcenter

Circumcenters are found at the intersection of the perpendicular
bisectors of the sides of the triangle.
The perpendicular bisector is created by bisecting
each side of the triangle with a perpendicular line. A circumcenter
is also the center of a circle, which passes through each of the three vertices
of the triangle.

Incenter

An incenter
is found at the intersection of the angle bisectors of a triangle. The incenter
is the center of the incircle, an inscribed circle of
the triangle.