Triangle Centers
The Circumcenter of a Triangle is the point equidistant from all three of its vertices. It is the
point where the center of a circle can be drawn that circumscribes the triangle. It is constructed
by finding the intersection of the the perpendicular bisectors of the sides Lines that contain the
same point are called concurrent. Concurrence is the concept of three or more lines intersecting in a single
(common) point, having a single point of intersection. Notice that the point of concurrence is not
necessarily in the interior of the triangles.
Perpendicular Bisectors of an acute triangle.
Perpendicular Bisectors of an obtuse triangle.
To construct the circumcenter you construct the perpendicular bisectors (fish-eye) for each side. The method
can be simplified if you use a radius larger than half of the longest side for each vertex as shown below
The Incenter of a Triangle is the point equidistant from all three of it’s sides. It is the point
where the center of a circle can be drawn that fits inside the triangle. It is constructed by finding
the intersection of the the angle bisectors of all three vertices
Lines that contain the same point are called concurrent. Concurrence is the concept of three or more
lines intersecting in a single (common) point, having a single point of intersection.
Notice that the point of concurrence is always in the interior of the triangles.
Angle Bisectors of an acute triangle.
Angle Bisectors of an obtuse triangle.
To construct the incenter you construct the angle bisectors for each vertex. The method can be clearer if you
draw the congruent arc (that make the “X”) outside of the triangle as shown below
To find the radius for the incenter circle pull a perpendicular from the incenter point to the longest side (as shown in
orange below)
Medians: A median of a triangle is a segment joining any vertex to the midpoint of the opposite
side. The medians of a triangle are concurrent. Notice that the point of concurrence is in the
interior of the triangles.
Medians of an acute triangle.
Medians of an obtuse triangle.
Archimedes showed that the point where the medians are concurrent is the center of gravity of a
triangular shape of uniform thickness and density. This point where the medians are concurrent
is called the centroid of a triangle. If you cut a triangle out of cardboard and balance it on a
pointed object, such as a pencil, the pencil will mark the location of the triangle's centroid.
The centroid divides the medians into a 2:1 ratio. The section of the median nearest the vertex
is twice as long as the section near the midpoint of the triangle's side.
Altitudes: An altitude of a triangle is a segment from any vertex perpendicular to the line
containing the opposite side. The lines containing the altitudes of a triangle are concurrent.
Notice that the point of concurrence is not necessarily inside the triangle.
Altitudes of an acute triangle.
Altitudes of an obtuse triangle.
The point where the lines containing the altitudes are concurrent is called the orthocenter of a
triangle. (The prefix "ortho" means "right".)
Euler Line: In any triangle, the circumcenter, centroid,
and orthocenter are collinear (lie on the same straight
line). The collinear line upon which these three points
lie is called the Euler line. The centroid is always
located between the circumcenter and the
orthocenter. The centroid is twice as close to the
circumcenter as to the orthocenter.
The word "Euler" is pronounced as if it were spelled "Oiler" and refers to the mathematician Leonhard
Euler (1707-1783).