This investigation starts with any triangle **ABC** and examines the
triangle formed by the points where the extended altitudes meet the circumcircle.
Then this triangle is compared to the orthic triangle.

Let's begin by constructing any triangle **ABC** and its extended
altitudes. Remember that an** altitude **is the perpendicular line drawn
from a vertex to the opposite side of a triangle. The intersection of the
three altitudes is called the **orthocenter** ; we label it **H**.

Let's connect the feet of the altitudes to construct triangle **DEF**.
This is the **orthic triangle**.

Now we need the **circumcircle**. First we find the **circumcenter
**of the triangle **ABC**. It is the point which is the same distance
from each vertex. To find it, we construct the perpendicular bisectors of
each side of the triangle; they are colored blue to distinguish them from
the extended altitudes. The bisectors meet at **Ci**.

Notice that the perpendiculaar bisectors are all parallel to the altitudes. Why? Now Ci is the same distance from each vertex and is the center of the circumcircle. Let's use the distance from Ci to C to be the radius of the circumcircle.

Now we'll hide the perpendicular bisectors to make viewing a little easier.
Notice that each extended altitude intersects the circumcircle in two points.
One is at the vertex; recall that each vertex is the distance of a radius
from the circumcenter. The other is the intersection we've been working
for to construct the final triangle, **RST**.

Triangle **RST** certainly appears to be similar, that is it has the
same shape and same size angles, as triangle **DEF**.

Click
here to see a GSP sketch which can be manipulated to show the relationships
of the triangle by moving **A**,**B**, or **C** along the circumcircle.