Centers of a triangle

(Assignment 4)

by

J. Matt Tumlin, Cara Haskins, Robin Kirkham, and Venki Ramachandran

Here we show that the distance between the Centroid, Circumcenter and the Orthocenter of any triangle always maintain a constant ratio. HG = 2GC.

We construct the three centers of the triangle first.

1.   The Centroid G is the intersection of the tree medians of a triangle.  The median being a line joining the midpoint of a side with the opposite vertex.

Triangle DEF has the Centroid at G.  Whatever shape the triangle assumes, G lies within the triangle.

2.   The Orthocenter of a triangle is the point of intersection of the altitudes dropped from each vertex to the opposite side.

The orthocenter could be inside or outside the triangle.

3. The circumcenter of a triangle is the point of intersection of the perpendicular bisectors of the sides.  A circle drawn with the circumcenter as the center would pass through the three vertices of the triangle.

The circumcenter C of a triangle may lie within or outside of the triangle as demonstrated above.  It can however be observed that the locus of C will be one of the three perpendicular bisectors.

3.   The INCENTER of a triangle is the point inside the triangle that is equidistant from the three sides. Since a point interior to an angle that is equidistant from the two sides of the angle lies on the angle bisector, then I must be on the angle bisector of each angle of the triangle.

Needless to say, irrespective of the shape of the triangle, the in-center lies inside the circle.  Further, a circle drawn with the In-center as a center and radius equal to the perpendicular distance to any one side will touch the other two sides.

5. To construct G, H, C and I for a given triangle and observe them

The circumcenter C, the Centroid G and the Orthocenter are always collinear.  When the triangle becomes an equilateral triangle, all these four points become concurrent.  Further, the ratio of the distances between C, G, and H is constant.

The circumcenter C, the Centroid G and the Orthocenter are always collinear.  When the triangle becomes an equilateral triangle, all these four points become concurrent.  Further, the ratio of the distances between C, G, and H is constant.

The Orthocenter, H, is the point of intersection of the perpendiculars BY, DZ, and QK.  The Circumcenter, C, is the point of intersection of EF, RS, and XL. These three lines are the perpendicular bisectors of the sides QD, BQ, and BD.  The Centroid, G, is the point of intersection of BX, QE, and DR. These are the lines drawn from the vertex to the midpoint of the sides opposite to the vertex.

We now need to prove that the length HG is twice the length of GC.

Consider the two triangles HQG and CGE.  These two triangles are similar.  Proof of this similarity is easily recognized by observing that lines FE and QK are parallel.  Thus angle ECG = angle QHG. Further, angle CGQ = angle HGE being vertically opposite angles.

Therefore, CG : GH as QG : GE.

We know that G being the Centroid of the triangle divides QE in the ration of 1:2. Thus, QG:GE = 1:2.

Therefore, CG : GH as 1 : 2.