**In this assignment
we will look at the perpindicular bisectors of the sides of any
triangle and see if we can prove that they are concurrent. In
order to understand the process and what we are trying to do,
one must first understand what the definition of a perpindicular
bisector is, as well as what we mean by concurrent.**

**First we all
know, as we learned in preschool or maybe even earlier that a
triangle is a shape that is made up of 3 sides. I am sure you
recall words, no matter how long you have been out of school,
such as "acute", obtuse", "right triangle",
and "isoceles" being used to describe and define triangles.
Most likely you cannot recall what is meant by an angle bisector
nor concurrence. I have defined both terms below, and attached
photos to give you a visual of each as well.**

**Perpendicular
bisector:****
**Perpendicular bisectors are lines running through the
midpoint of each side in a triangle at 90 degree angles

**Concurrent:**** **In geometry,
three or more lines are said to be **concurrent** if they intersect
at a single point.

**Now let's investigate a triangle and see if the perpendicular
bisectors of the sides are concurrent.........**

First let's construct our triangle and one of the perpendicular bisectors.

We see in the picture above that our dashed red line is perpendicular to and intersects segment AB at the midpoint. You also discovered from the link above, that as you move point A or B the perpendicular bisector moves as well, so as our triangle changes shape our perpendicular bisector adjusts.

We will now add our other two perpendicular bisectors and see if (1) the are concurrent, and (2) as we change the shape of our triangle if there remains a common point of intersection and (3) what happens to this point as we change the shape?????

We see that there is in fact a point of intersection for all
three perpendicular bisectors, therefore they are **concurrent**.
This point acutally has a geometric name, it is labeled the circumcenter
of the triangle and is often labeled with the letter ** C **as we have done in our
picture above. This Point is labeled the circumcenter, which is
defined as.....The

Now let us explore our triangle and see as we manipulate the shape of our triangle what happens to the circumcenter? (click on the link and then move any of the veritices of the triangle in the picture that appears and see how the orthocenter changes) CLICK HERE

We see that as we manipulate our triangle we always have a circumcenter, but it is not always contained in the circle. In fact you may notice that as one of the angle becomes obtuse, greater that 90 degrees, the circumcenter moves outside the circle.