Assignment I

Math 7200

An Exploration on Center of Triangles: Circumcenter (C)

by

Samuel Obara

The circumcenter (C) of a Triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies in the perpendicular bisector of the segment determined by the two points, C is on the perpendicular bisector of each side of the triangle.

Construction

Draw any triangle ABC as shown below.

Then construct perpendicular bisector segment AB, BC and AC. It can be noted that they seem to meet at a common point (C) as shown in figure 1. It seems that (C) lies outside the triangle in this case! Let us observe what happens in the following three cases.

 

 

 

 

 

 

From the three cases stated it seems that when in case 1 the triangle is abtuse and the circumcenter seems to be outside the triangle. For case 2 i.e. acute triangle, C seems to be inside the triangle. One interesting case is when the triangle is right i.e. case 3. In this case the center C seems to lie in the midpoint of the hypotenuse. For more illustration, click the animation on the gsp file in this folder. As it was earlier stated that C is a point, which is equidistant, let us see what happens.

As noted the circle seems to pass through the three vertices. Try to play with the centroidcircum-1gsp file by clicking the animation in this folder. It will be noted that the circle always passes through the three vertices.

Proof of the concurrence of the perpendicular bisector

By referring to figure 5, let the perpendicular bisector of AB and BD meet at C. Construct a line segment from C to AD such that CM is perpendicular to AD. Triangle APC and BPC are congruent hence AC = BC, also triangle BNC and DNC are congruent hence BC = DC and hence AC = DC. Therefore we can show that triangle AFC and DFC are congruent and hence AF = FD showing that line FC is the perpendicular bisector of side AD. There AC = BC =DC hence any circle with C as the center and passing through on of the vertices will pass through all of them. So in general the only triangle where both G and C will lie at the same point is when you have equilateral triangle.