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 incircle, 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.
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.
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.