By Brandie Thrasher
LetŐs investigate altitudes and orthocenters, and begin with triangle ABC with its constructed orthocenter H.
Now, lets contsrtuct the orthocenter for triangle HBC (labeled G), Followed by the orthocenter of triangles HAB (labeled I), and HAC (labeled J)
Now we will construct the circumcircles of triangles ABC, HBC, HAB and HAC
Once the construction is complete, our orthocenterŐs and circumcircles made a beautiful display
But what exactly do we have? Lets construct some conjectures:
1. The radii of the circumcircles constructed from triangle ABC form a hexagon.
2. These same radii form three triangles
3. The three newly formed triangles are similar to triangles HAB, HBC, and HAC.
4. The area of all four circles is congruent
Students can take these conjectures and turn these into proofs, investigating the many interesting ways altitudes and orthocenterŐs affect one another.