Orthocentric Powers

by

Christa Marie Nathe

In this exploration we are going to construct several orthocenters of inter-related triangles and observe some relationships between their orthocenters.  In addition we will also construct circumcircles that are inter-related to both each other and the triangles used to construct them.  Recall that an orthocenter of a triangle is formed from the intersection of the three lines of the altitude of the triangle. An altitude is a perpendicular segment from a vertex to the line of the triangle on the opposite side.

 

Lets begin with constructing our basic triangle, which we will label ABC

 

The orthocenter of ABC is point H.

 

 

Now we have four triangles.

ABC

HBC

HAB

HAC

We will now construct the orthocenters of each of the aforementioned four triangles.

 

 

Orthocenter of HBC is point J, which is also the vertex A of our original triangle.

 

Orthocenter of HAB is point L and consequently vertex point C of our base ABC triangle.

 

The orthocenter of HAC is point M, as well as the vertex B of the triangle ABC.

Next we will construct the circumcircles for each triangle

ABC

HBC

HAB

HAC

 

Remember that circumcircles are constructed via circumcenter point of a triangle. The circumcenter of a triangle is the point that is equidistant from the three vertices of the triangle. By taking the perpendicular bisector of each of the segments between two vertices of the triangle and their intersection it will yield the circumcenter. The circumcircle is the circle created by the vertices of the triangle, with the circumcenter point as its radius.

 

Circumcircle of ABC

 

 

Circumcenter of HBC

 

 

 

Circumcenter of HAB

 

Circumcircle of HAC

Combining all of our constructions of the orthocenters for the four triangles, the circumcircles and connecting the circumcenters we obtain the following figure.

 

 

Now that we have made our composite construction of all of elements, lets see what happens when the vertex A is moved to where its orthocenter is, that is at point H. As a result, the pink circle from triangle HBC is swapped with the green circle from ABC since changing the position of A altered its circumcenter. As H moved, the triangle HBC became larger thus causing its circumcenter and circumcircle to shift.

 

 

 

 

Here, point H is exchanged with vertex B. Can you anticipate which circle will be shifted?

If you expected the green circumcircle from triangle ABC and the blue circumcircle from triangle HAC to be swapped then you were correct!

 

Finally, H is switch with vertex C and we derive the following construct.

                                                                                              

Yes, the green circle of the ABC triangle and the turquoise circle of the HAB triangle were switched.