Definition: The orthocenter is the intersection of the altitudes of triangle ABC.
After constructing triangle ABC, it is possible
to construct the orthocenter and label it H. Construction of triangles
HBC, HAB, and HAC can then made.
Upon construction of the orthocenter for triangle HBC, I noticed that it coincided with vertex A of my original triangle. I decided to drag a vertex of triangle ABC to see if this held true and it did. As I constructed the next two orthocenters, one for triangle HAB and one for triangle HAC, I confirmed the guess that these orthocenters would fall on the vertices C and B, respectively. Therefore, when constructing orthocenters of this nature, they will coincide with the vertex of the original triangle that is not being used in the construction.
Definition: The circumcircle of a triangle is constructed from the circumcenter of a triangle, which is the intersection of the perpendicular bisectors of a triangle.
Next, I began to construct the circumcircles of all the above triangles.
After completing this, two things seemed to stand out. One was the petal-like object formed by the overlapping of the circumcircles. The second, which was my focus, was the hexagon formed from the points of the circumcenters and orthocenters.
Triangle ABC was an acute triangle and the outer figure formed a hexagon. By dragging the vertex of the triangle to obtain an equilateral triangle, the hexagon began to look regular (all sides, all angles congruent). I confirmed this by making measurements.
Therefore, the hexagon formed by the orthocenters and circumcenters from triangle ABC, is regular when triangle ABC is an equilateral.
Next, a test of other triangles. . .
When ABC triangle is right, the hexagon forms a rectangle.
When triangle ABC is isosceles right triangle, the hexagon forms a
When triangle ABC becomes an
obtuse triangle, the figure is no longer a closed polygon.