Centers of a Triangle

Assignment #4: Centers of a Triangle

Problem 12: Prove that the three perpendicular bisectors of the sides of a triangle are concurrent.

Laura Singletary

A perpendicular bisector of a line segment, call it AB, is a line perpendicular to AB and passes through the midpoint of AB. Additionally, any point on the perpendicular bisector of AB is equidistant to A and B.

The circumcircle is a triangle's circumscribed circle. It is the unique circle that passes through each of the triangle's three vertices. The center of the circumcircle is the circumcenter of the triangle. The circumcenter is formed by concurrency of the triangle's three perpendicular bistectors.

Let us prove that the perpendicular bisectors of a triangle are concurrent.

Proof:

Remember that the perpendicular bisector of any line segment AB is the set of points that are equidistant from A and B.

Given triangle ABC, construct the midpoint of segment AB calling it MAB. Construct the line perpendicular to AB through the midpoint, calling it line j - the perpendicular bisector of AB. Similarly, construct the midpoint of segment AC calling it MAC. Construct the line perpendicular to AC though the midpoint, calling it line k - the perpendicular bisector of AC. Construct the intersection of the perpendicular bisectors j and k, call the intersection O.

Knowing that the perpendicular bisector j is the collection of points equidistant to B and A, then we can say that BO= AO.

Similarly, knowing that the perpendicular bisector k is the collection of points equidistant to A and C, then we can say AO = CO.

Then, using the transitive property of equality and the fact that BO=AO=CO, we can say that BO=CO.

By saying that BO=CO, then O is equidistant from B and C. Therefore, O must reside in the collection of points know as the perpendicular bisector of BC.

Therefore, since O is a point on the perpendicular bisector of AB, AC, and BC, then we can say that the perpendicular bisectors of any given triangle are concurrent. Q.E.D.

Below is an example of the concurrency of the perpendicular bisectors of triangle ABC.

Additionally, it is interesting to note that the concurrency of the perpendicular bisectors of triangle ABC form the point O which is also known as the circumcenter of the circumcircle of the triangle. Using O as the circurcenter, we can construct the circumcircle that passes through all three vertices of the triangle ABC, as seen below.