Proving the concurrency of

the three perpendicular bisectors

of the sides of a triangle.

Write-up by

Blair T. Dietrich

EMAT 6680

Assignment #4 Problem #12

Given triangle ABC, prove that the three perpendicular bisectors of the sides of the triangle are concurrent.

Proof:

Construct the perpendicular bisectors of and , namely and , intersecting at point I.

The point of concurrency I can be shown to be equidistant from points A and C:

Angle AGI and angle CGI are right angles since .

andare right triangles by the definition of a right triangle.

since G is the midpoint of .

by the reflexive property.

by the Leg-Leg Theorem for right triangles.

since corresponding parts of congruent triangles are congruent.

Therefore, *AI
*= *CI* by the definition of congruent segments

Similarly, it can be shown that and that I is equidistant from points B and C.

It follows then that, since and , by the Transitive Property.

We now must show that the point I lies on the perpendicular bisector of :

Construct a line through I that is perpendicular to . Label the point of intersection J.

Angle AJI and angle BJI are right angles by the definition of perpendicular lines.

andare right triangles by the definition of right triangles.

It has been shown that the three perpendicular bisectors are concurrent at point I.

Note that *this* point I is not necessarily on the
interior of triangle ABC, nor does it necessarily lie on the angle bisectors of
triangle ABC; therefore, this point I is not, in general, the incenter of the
triangle.

Another example showing that I is not the incenter.

However, in each case the perpendicular bisectors are concurrent.