Exploring Orthocenters
Rita Meyers

In this exploration we will look at an orthocenter of a given triangle and some different investigations of the orthocenter.

Let's start with a given triangle ABC and it's orthocenter H.

Now let's construct triangles HAB, HBC, and HCA

We'll first explore these new triangles by contructing the orthocenters of these three triangles

H' = Orthocenter of HAB

H'' = Orthocenter of HBC

H''' = Orthocenter of HCA

Looking at the orthocenters of these three triangles we see that each one correlates to a vertex of the given triangle.

Next we'll explore further, by looking at the circumcircles

First we will construct the circumcenter of the given triangle in order to find the circumcircle.

Now let's construct the circumcircles of triangles HAB, HBC, and HCA

Circumcircle of HAB
Circumcircle of HBC
Circumcircle of HCA

This is what we have when we have the circumcircles of all four triangles.

Let's explore this sketch
What happens when we make the orthocenter (H) of the original triangle (ABC) equal to a vertex of the triangle ABC (or an orthocenter of triangles HAB, HBC, or HCA)?

If we move vertex B so that it overlays the orthocenter H this is what we see

It appears that the circumcircle of triangle ABC is now equal to the circumcircle of triangle HCA

Does this appear to be the same when the orthocenter is equal the other vertices of the original triangle

H = Vertex A
H = Vertex C

If fact it is true that the circumcircle of the original triangle is equal to one of the circumcircles of the triangles constructed with the orthocenter H, when the orthocenter H is equal to one of the vertices of triangle ABC

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