Assignment 4:

Centers of a Triangle

The purpose of this assignment is to investigate the construction and properties of several different centers of a triangle - the centroid, the orthocenter, and the circumcenter.  Also, we will investigate the relationship between the three centers.

Centroid

Construction: The centroid is found by finding all three midpoints of the segments that make up the triangle and connect each midpoint to the corresponding vertex opposite each midpoint.  After completing all three lines from each midpoint to the opposite vertices, you can see that all lines go through the same point in the center of the triangle.  This point is the intersection point of the lines.  Once the intersection point is constructed, it becomes the centroid of the triangle.  Click here to see the GSP construction of the centroid.

Properties:  The centroid is one center of the triangle, but it is not equal distance from each of the three vertices.  No matter which way you move or change the triangle that you have constructed along with the centroid, the centroid is always in the center of the triangle.

Circumcenter

Construction:  The circumcenter of a triangle is found by finding the midpoints of the segments that comprise a triangle and drawing the perpendicular bisectors of each of the three segments.  After completing tall the perpendicular bisectors of all three segments, you can see that the perpendicular bisectors intersect each other at one point.  Further, this point is equal distance from all three vertices of the triangle.  By constructing a circle using the intersection point we have found as the center and passing through one of the vertices of the original triangle (circle by center + point), we now have a circle in which all the vertices of the original triangle lie on the circle.  Further, the center of the constructed circle is the circumcenter.  Click here to see the GSP construction of the circumcenter.

Properties:  When looking at the circumcenter of an equilateral or acute triangle, the circumcenter is in the center of the triangle (click here to view the circumcenter of an acute and equilateral triangle).  However, when the triangle is obtuse, the circumcenter is outside of the triangle (click here to view the circumcenter of an obtuse triangle).  When looking at a right triangle, the circumcenter is on the midpoint of the hypotenuse of the triangle (click here to view the circumcenter of a right triangle).  When moving a given triangle from an acute angle to an obtuse angle, the circumcenter exits through the midpoint of the segment opposite the obtuse angle.

Orthocenter

Construction:  The orthocenter is found by constructing a line perpendicular to a vertex and the segment of the triangle that is opposite that vertex.  This process is repeated for the other two vertices and their corresponding opposite segments of the triangle. This will construct three lines with a common intersection point.  This intersection point is the orthocenter.  Click here to see the GSP construction of the orthocenter.

Properties:  When looking at a right triangle, the orthocenter is at the vertex of the angle opposite the hypotenuse (click here to see the orthocenter of a right triangle).  When looking at an equilateral triangle, the orthocenter is in the center of the triangle (click here to see the orthocenter of an equilateral triangle).  When looking at a triangle with an acute angle, the orthocenter is inside the triangle (click here to see the orthocenter of an acute triangle).  When looking at a triangle with an obtuse angle, the orthocenter exits the triangle through the vertex forming the obtuse angle and then through the vertex point (click here to see the orthocenter of an obtuse angle).

Observing All the Centers Together

Construction:  Click here to see the GSP construction of all the triangle centers on the same triangle.

Properties:  All three of the centers are the same when the triangle is equilateral (click here to see the way all the centers are exactly the same in an equilateral triangle).  Also, the centroid is always in the center of the orthocenter and the circumcenter no matter how you change the triangle.  When studying the distance between the orthocenter and the circumcenter, the distance between the circumcenter and the centroid is a ratio of 1/3 and the ratio of the distance between the orthocenter and the centroid is 2/3.