The Department of Mathematics Education


The Investigation of the Circumcenter of a triangle

by GooYeon Kim


There is a triangle given. The circumcenter (C) of a triangle is the point in the plane equidistant from the three vertices of the triangle. Then we can get the circumcircle, the circumscribed circle, of the triangle.

 

 

Now, let's think about where the circumcenter is. Let's suppose the circumcenter of a triangle lies on the perpendicular bisector lines of each side of a triangle. Then we should prove this conjecture.

We know, first, a segment AC is the radius of the circumcircle. Furthermore, AC=BC=DC=radius by the definition of the circumcenter. Then we can get a triangle CBD which is a isoceles.

We certainly know the fact that the point, circumcenter, C lies on the line of perpendicular midpoint of a segment BD below picture.

As a result, we proved our conjecture. Then, can we say that C is the intersection point of the three perpendicular bisector lines?

Next, we are exploring that the three perpendicular bisector lines of each side meet only at one point which is the circumcenter of the triangle. Can you imagine what will happen in your head? If so, keep it while we are going through, comparing yours with following pictures and processes.

Take two perpendicular bisector lines first and locate the intersect point of them. Before looking at it, draw what 's like in your head. Now, see

Then, in order to justify the conjecture, draw the perpendicular, not bisector, line from the point C to a segment AD. Let put the intersect point of them label L. We already know CA= CB and CB = CD, which results in CA=CD. Consequently, a triangle CAD is isoceles. Then we have AL = DL. That means segment CL is a part of the perpendicular bisector line of AD from the point C. So the perpendicular bisector lines of each side of a triangle meet at only one point that is called the circumceter. Finally, we proved our conjecture.

Now, we are investigating the circumcenter's location for various shapes of triangles. Click here.

 


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