In this investigation, we are going to show
that the lines of the three altitudes of a triangle are concurrent
and that the three perpendicular bisectors are concurrent. This
means that all three altitudes have a common point of intersection
and all three perpendicular bisectors have a common point of intersection.
First, let's define the terms ** altitude** and

The ** altitude** of a triangle is
a perpendicular segment from a vertex to the line of the opposite
side. We will note that the foot of the perpendicular may be the
on the extension of the side of the triangle. Let's show an illustration
of the three altitudes.

As we can see from the graphic representation, it appears that the altitudes do intersect at a common point G and are therefore concurrent. What if we change the shape of the triangle? Will the three altitudes still be concurrent? Click HERE to change the shape of the triangle and observe that the altitudes are always concurrent.

The ** perpendicular bisector** is
a perpendicular line through the midpoint of one of the sides
of the triangle. Let's show an illustration of the three perpendicular
bisectors.

As we can see from the graphic representation, it appears that the perpendicular bisectors do intersect at a common point G and are therefore concurrent. What if we change the shape of the triangle? Will the three perpendicular bisectors still be concurrent? Click HERE to change the shape of the triangle and observe that the perpendicular bisectors are always concurrent.

One thing you should notice is that the points
of concurrency do not always lie inside the triangle. When the
triangle is obtuse, the points of concurrency lie outside the
triangle. The common intersection of the three lines containing
the altitudes is called the ** orthocenter** of a triangle.
The common intersection of the three lines containing the perpendicular
bisectors is called the

If you draw a circle using the circumcenter
of the triangle as the center and the distance between the circumcenter
and a vertex of the triangle as the radius, you get a circle that
passes through each of the vertices of the triangle. This is because
the circumcenter is equidistant from the three vertices of the
triangle. This circle is called the ** circumcircle**.
Here is an illustration of the circumcircle of a triangle.

Click HERE to change the shape of the triangle and observe how the circumcircle changes.

If we connect the midpoints of the triangle
from above, we get another triangle. This triangle is called the
** medial** triangle of our original triangle. Below
is an illustration of the medial triangle.

You may notice that the perpendicular bisectors of the original triangle are the altitudes of the medial triangle. Click HERE to change the shape of the triangles and observe changes.