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 of the given side.

Move any vertex to see how the Centroid moves.

A centroid will remain inside the triangle for the three types of triangles categorized by their angles, as shown below.

Acute Triangle Right Triangle

Obtuse Triangle

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.

Move any vertex to explore the Orthocenter.

Right Triangle Obtuse Triangle

Acute Triangle

You will notice that orthocenters, unlike centroids, can be inside, on or outside the triangle.

Further exploration...

What path does the orthocenter follow when a vertex is animated along a line parallel to the base?

#### Orthocenter with Animation

What path is followed by the trace of the Orthocenter?

Orthocenter with Traces

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.

Move any vertex to explore the Circumcenter

Centers of Circles, EMAT 6680 Nored

#### Circumcenter

Acute Triangle    Right Triangle

Obtuse 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. Any thoughts on finding the radius of the incenter?

Move a vertex to explore the Incenter. There are some limitations to this construction due to issues with Java and Sketchpad. Incenter

Right Triangle        Acute Triangle

Obtuse Triangle

Notice that the intersection of the triangle and the circle only occurs on the angle bisectors for an equilateral triangle. Recall that an equilateral triangle is equiangular also. To find the radius of the incircle drop a perpendicular line from a side to the incenter. This distance is equal from any side.