# Perpendicular Bisector of a Triangle

The **perpendicular bisector** of a segment is a line perpendicular to the segment that passes through its midpoint. It has the property that each of its points is equidistant from the segment’s endpoints.

A **perpendicular bisector** of a triangle *ABC* is a line passing through the midpoint *M* of each side which is perpendicular to the given side. For example, the perpendicular bisector of side *a* is M_{a}.

There are three perpendicular bisectors in a triangle: M_{a}, M_{b} and M_{c}. Each one related to its corresponding side: *a*, *b*, and *c*.

These three perpendicular bisectors of the sides of a triangle meet in a single point, called the **circumcenter**.

The circumcenter is the center of the triangle’s circumscribed circle, or **circumcircle**, since it is equidistant from its three vertices.

The radius (*R*) of the **circumcircle** is given by the formula:

The relationship between the radius *R* of the circumcircle, whose center is the circumcenter *O*, and the inradius *r*, whose center is the incenter *I*, can be expressed as follows:

Download this **calculator** to get the results of the formulas on this page. Choose the initial data and enter it in the upper left box. For results, press ENTER.

Triangle-total.rar or Triangle-total.exe

Note. Courtesy of the author: **José María Pareja Marcano**. Chemist. Seville, Spain.

## How do you draw the perpendicular bisectors of a triangle?

The **perpendicular bisectors** of a triangle can be easily drawn with a ruler and compass.

Let’s start by drawing the perpendicular bisector of a line segment *S*, whose ends are *X* and *Y*:

- Open the compass more than half of the distance between
*X*and*Y*, and draw arcs of the same radius centered at*X*and*Y*. - There are two points where these two arcs meet. We’ll call them point
*P*, and point*Q*. - Place the ruler where the arcs cross (i.e. connect point
*P*and point*Q*), and draw the line segment*M*, which is the perpendicular bisector of the line segment*S*.

Now, we are going to proceed in the same way on all three sides of the triangle to draw its three perpendicular bisectors (M_{a}, M_{b} and M_{c}):