Perpendicular Bisector of a Triangle

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Drawing of the three perpendicular bisectors of a triangle and the circumcenter

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 Ma.

Drawing of the three bisectors of a triangle, the circumcenter and the circumscribed circumference

There are three perpendicular bisectors in a triangle: Ma, Mb and Mc. 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:

Formula for the radius of the circumscribed circle in the triangle

Drawing of circumscribed and inscribed circles for the relationship between their radius

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:

Formula for the relationship between the radius of the circumscribed and inscribed circle in a triangle

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:

  1. 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.
  2. There are two points where these two arcs meet. We’ll call them point P, and point Q.
  3. 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.
Drawing of the perpendicular bisector of a segment

Now, we are going to proceed in the same way on all three sides of the triangle to draw its three perpendicular bisectors (Ma, Mb and Mc):

Drawing of the bisectors of a triangle using a compass and ruler

Lines Associated with a Triangle

AUTHOR: Bernat Requena Serra

YEAR: 2020


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