**1. Construct any triangle ABC.**

**2. Construct the Orthocenter H of triangle
ABC.**

**3. Construct the Orthocenter of triangle
HBC.**

**4. Construct the Orthocenter of triangle
HAB.**

**5. Construct the Orthocenter of triangle
HAC.**

**6. Construct the Circumcircles of triangles
ABC, HBC, HAB, and HAC.**

**Circumcircle of triangle ABC:**

**Circumcircle of triangle HBC:**

**Circumcircle of triangle HAB:**

**Circumcircle of triangle HAC:**

**7. Conjectures? Proof?**

**I could not find anything that really
jumped out at me when I did the orthocenters and circumcircles
of these triangles. I can conjecture though that triangles that
are named are always within the the circumcircle, hence the name...circumcircle.
I also noticed that by scrolling from one circle to the next,
I found that as the circumcircle was made on triangle ABC, segment
BC was highlighted in RED. When I observed the circumcircle of triangle
HBC, I found that segments HB and HC were highlighted in RED. This same condition
was also true for triangle HAB, where segments HA and HB were
highlighted in RED;
also for triangle HAC, where segments HA and HC were highlighted
in RED.
What does this say for the constructions or the circumcircles
or even the orthocenters. I do not know the answer to this one.
Maybe you can e-mail me with some information about it. It would certainly be helpful to my
students...and of course their loving teacher!!!
If you continue on, I do believe I have stumbled onto something
quite interesting!!!**

**8. Construct the nine point circles for
triangles ABC, HBC, HAC, and HAB.**

**Nine Point Circle for triangle ABC:**

**Nine Point Circle for triangle HBC:**

**Nine Point Circle for triangle HAC:**

**Nine Point Circle for triangle HAB:**

**9. Construct triangle ABC, its incircle,
its three excircles, and its nine-point circle. Conjecture?**

**Look at the dashed lines in each figure
below. What do you see in each figure? Are there any major changes
from one phase to the next? Also, look at each shape of the triangle
being presented. What shapes are there? What sort of conjecture
can you come up with about the triangles and the figures made
by the dashed lines? How were the figures made? Hint: Construct
problems 1-6, 8, &9. You may do this now by clicking:**

**here**** for the
script of an orthocenter
(don't forget to choose 3 points first and follow the directions
above)!**

**here**** for the
script of circumcircles**

**here**** for the
script of the nine point circles**

**here**** for the
script of the incircle**

**here**** for the
script of an excircle**

**What can be said for all
these discoveries?**