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Centers of a Triangle

(Assignment 4)

by

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

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Here we show that the distance between the CENTROID, CIRCUMCENTER and the ORTHOCENTER of any triangle always maintain a constant ratio. HG = 2GC

(key; blue dot = H; purple dot G; green dot C . HG green line; GC orange line)

 H               G                                                                  C

We construct the three centers of the triangle first. The Orthocenter is the point of intersection of the perpendiculars BH and CK.  The circumcenter is the point of intersection of EF and GL. These two lines are the perpendicular bisectors of the sides CD and BD.  The Centroid is the point of intersection of BG and CE. 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 HCG and CAE.  These two triangles are similar.  Proof of this similarity is easily recognized by observing that lines FE and CK are parallel.  Thus angle ECG = angle H. Further, angle CGA = angle HGC 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 2:3. Thus, QG:GE = 1:2.

Therefore, CG : GH as 1 : 2.

If you think this was funÉ

Continued investigations of triangles:

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.

 D                                                                                             G                           E                                                                                                                              F

Triangle DEF has the Centroid at G  (orange dot in center).  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.

 I                                                                                                                                                                      J         K

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.

 H                                                                                                         H

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.

4.  The INCENTER of a triangle is the point inside the triangle that is equidistant from the three sides.  Such a point would obviously lie on the angle bisector of the vertices.

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

 H                                                              G              I                                                                                         C

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.

6. The INCENTER (I) of a triangle is the point on the interior of the triangle that is equidistant from the three side.  Since a point interior to an anle that is equidistant from the two sides of the angle lies on the angle bisetor, ten I must be on the anlge bisector of each angle of the triangle.

 I

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