Charlie Conway 7/30/2009

Triangle Centers

Incenter - The intersection of the angle bisectors of the three angles of the triangle. Also the center of the triangle's incircle.

Circumcenter - The intersection of the perpendicular bisectors of the three sides of the triangle. Also the center of the circumcircle.

Orthocenter - The intersection of the triangle's altitudes.

Centroid - The intersection of the three medians of the triangle. Also the center of gravity of the triangle.

Euler Line - The line containing the circumcenter, orthocenter, and centroid. These points are always collinear.

GSP sketches of the centers: Individual Centers or All Centers on One Triangle

Investigations

When we manipulate the triangle with all four centers shown, we can discover some interesting properties.

We already know that the circumcenter, orthocenter, and centroid are always collinear., contained by the Euler Line. However, by manipulating the lengths of the sides of the triangle, we discover that the incenter is also collinear. when the triangle is isosceles. The Euler Line also goes through the vertex between the two sides of the same length, as well as the midpoint of the opposite side of an isosceles triangle. Since the line goes through a vertex and the incenter, by the definition of the incenter, this line must bisect the angle of that vertex.

When we manipulate the sides of the triangle to create an equilateral triangle, we can see that all of the centers of the triangle not only lie on the same line, but lie on top of each other. They are the concurrent. This is rather simple to prove. In an equilateral triangle the perpendicular bisector of a side goes through the opposite vertex. This segment is also the altitude, perpendicular to a side through the opposite vertex. It is also the median, from the midpoint of a side to the opposite vertex. Since the perpendicular bisector divides the side of the triangle into two equal lengths and goes through the opposite vertex, by SAS it creates two congruent triangles, which means that this segment bisects the angle of the vertex, showing that the incenter is also concurrent.

The centroid has the property that the length of the segment from the vertex to the centroid is twice the length of the length of the segment from the centroid to the opposite side. Using this property and focusing on the incenter and the circumcenter, we can see that the radius of the circumcircle (from the circumcenter to a vertex) is twice that of the radius of the incircle (from the incenter to the opposite side), since they both lie on the centroid. Because the radius of the circumcircle is twice the radius of the incircle, the area of the circumcircle is four times as much as the area of the incircle.