# How to Find the Circumcircle of a Triangle

Instructor: Bob Bruner

Bob is a software professional with 24 years in the industry. He has a bachelor's degree in Geology, and also has extensive experience in the Oil and Gas industry.

Any triangle can be enclosed by one unique circle that touches each triangle vertex. This lesson will show how to easily construct the circumcircle of a triangle.

## Drone Racing Anyone?

Suppose you bought a new, speedy drone, and after a bit of practice wanted to enter some racing competitions. One of the first challenges you might encounter would simply be to race around three points on the ground. Your first tactic might be to draw a line between each point, giving you a triangle, resulting in the shortest distance you would have to race. But, in a race, this pattern would entail slowing down drastically to make the turn at each vertex. Another tactic might be to race around these points at a constant speed in a circular pattern that just touches each of the vertices. What you have done in this case is to create a circumcircle of a triangle. Let's find out how we can create the circumcircle of any triangular shape we are given.

## Constructing the Circumcircle of a Triangle

A triangle is guaranteed to have only one circle that contains each of its three corners. We can construct that circle if we can determine both the center of the circle and its radius. The center of the circle that circumscribes the triangle is called the circumcenter, and can be found using the following graphical method.

First, find the midpoint of any side of the triangle. Construct a new line extending into the triangle from this bisecting point that is perpendicular to the side of the triangle. Note that any point along this new perpendicular line has a similar property to the point that bisects the triangle segment. It is equidistant to both of the endpoints on that side of the triangle.

Now, take either of the remaining sides of the triangle and follow the same operation. First, find the bisector of the line segment that forms a side of the triangle, and then extend a perpendicular into the triangle from this point. Since no sides of a triangle can ever be parallel, these two new perpendicular bisecting lines can also never be parallel, and will always meet at one unique point.

Why doesn't it matter which sides we choose to use? The proof lies in the fact that each pair of vertices on the triangle is equidistant from any point along its perpendicular bisectors. Since all three points were used in the construction, the intersecting point any two bisectors have in common must be equidistant from all three points of the triangle.

If you want to do this operation again for the third side, you will see that the perpendicular meets at the same location, but this is not necessary. This new point we have found is, in fact, the circumcenter of the triangle. Measuring from the circumcenter to each apex point of the triangle gives the exact same distance, which by definition will form the radius of a circle. As can be expected, the radius of our circumcircle is also referred to as the circumradius. By using a circular drawing device such as a protractor, the circumcenter and circumradius can easily be used to draw the circumcircle of the triangle.

Each of these elements can be seen in the accompanying diagram. Each side of the triangle is bisected, and the red perpendiculars meet at point O, the circumcenter. The circumradius, R, is equidistant from each vertex of the triangle.

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