We will now explore the relationship between the orthocenters of a triangle and the circumcircles of the same triangle. We will also see how they interact with each other when the triangle is manipulated.
First let’s construct the triangle. We will create a random triangle and then find the orthocenter of that triangle. The orthocenter is found when a line is drawn from the vertex of the triangle to a point perpendicular to it on the opposite side of the triangle. You are able to see this in the picture below. Click here (ortho1.gsp) to change the shape of the triangle and see how the orthocenter moves.
We want to form three more triangles by connecting
each vertex to the orthocenter with a line segment. Now the orthocenters
of these new triangles will be found. Interestingly enough the orthocenters
of the triangles are the vertices of the main triangle. The orthocenter
of the triangle formed by line AB and vertices H1, H, and H2 is H3. The
orthocenter is not necessarily found within the triangle. The perpendicular
intersection may be found as if the sides of the triangle are extended
beyond the vertices. Thus the vertices of the original triangle are the
orthocenters of the smaller triangles formed from the original orthocenter.
Click here (Ortho2.gsp) to move the triangle below
around and explore how the orthocenters move and change.
Now we will create the circumcenters and circumcircles
of all the triangles. We will first find the circumcenter and circumcircle
of the main triangle. The circumcenter is the point that is equidistance
from the three vertices of the triangle. The center will leave the interior
of the triangle but will always pass through one of the sides’ mid-point.
Click (circum1.gsp) here to manipulate the circumcenter
Do you think the circumcircles and circumcenters
of the secondary triangles are found in the interior of their respective
triangles or on the exterior? Let’s create them and see.
The above picture shows an equilateral triangle. Notice that the circumcenters for the secondary triangles sit on the circumcircle of the primary triangle. Also the circumcenter and the orthocenter are the same.
There are countless explorations to be made with
this set of triangles and circles. Notice below that if you shift the
triangle around so that two of the circles over lap the triangle becomes
a right triangle and the secondary triangles disappear. We could certainly
find more things to explore but paper and time create limits. Click here
(circum3.gsp) to play with these figures further.