Alrighty…first, I constructed triangle ABC. Next, I needed to construct the orthocenter. However, in order to construct this thing called orthocenter, I had to brush up on my geometric definitions. Pretty much, the orthocenter is the intersection of the three altitudes of a triangle. What is an altitude? Well, an altitude is the height of a triangle; it is a line that is perpendicular to a side that also goes through the vertex opposite of that side. That’s a mouth full. Perhaps a visual will help.

Okay, the red line is the altitude perpendicular to side BC that goes through vertex A. The green line is the altitude perpendicular to side AC that goes through vertex B. And the purple line is the altitude perpendicular to side AB that goes through vertex C. That helps a lot…I’m a visual person. And good ol’ H is the orthocenter of triangle ABC. Why? Cuz it’s the intersection of the three altitudes. Oh yeah!

On to the next task…constructing the orthocenter of triangle HBC. Pretty much, I just repeated the steps above, and here is what I got:

It appears that the orthocenter of triangle HBC is vertex A from the original equation. I wonder if the orthocenters of other two triangles will be vertices of the original. Let’s find out:

Here is the diagram of the orthocenter of triangle HAB. It shows that the orthocenter is vertex C from the original triangle:

I bet that the orthocenter of triangle HAC will be vertex B from the original triangle…I like my odds…let’s see if I win:

Yep! The orthocenter is B.

All together:

Visually, it looks like too much going on…I guess I could’ve made the colors correspond to the graphs above…perhaps I’ll come back and tweak that. I’ll also come back to construct the circumcircles and make a conjecture…hopefully.

I’m back…on to the circumcircles. A circumcircle is a circle that passes through all three points of a triangle. The center of this circumcircle is called the circumcenter. The circumcenter is the point of intersection of the three perpendicular bisectors of the segments on a triangle. I read up on how to construct a circumcirle, and the following are diagrams of the circumcircle of triangles ABC, HBC, HAB, and HAC.

ABC:

HBC:

HAB:

HAC:

And here they are all together:

Alright, my conjecture about the orthocenters is this: when you take a triangle and construct it’s orthocenter, then use that point to serve as a vertex in the newly formed three triangles, and take the orthocenters of those triangles, then those orthocenters will be the vertexes of the orginal triangle. Goodness…that truly was a mouth full!

Conjecture about the circumcircles: well, it seems like the circumcirles of the newly formed triangle all go through the orthocenter, H, of the original triangle ABC. I guess this makes sense. This was pretty interesting to do.