Altitudes
and Orthocenters

by Kasey Nored

We constructed a triangle ABC with
the Orthocenter H, then constructed the Orthocenter of triangle HBC and the
Orthocenter of triangle HAB, and Orthocenter of triangle HAC. We also
constructed the Circumcircles of triangles ABC, HBC, HAB, and HAC. The image is
below.

To futher explore we
move a vertex of ABC to H and investigate the result.

Our
four circumcircles become three regardless of which vertex is translated to H.

The
circumcircle of ABC overlaps the circumcircle of H and the two vertices of the
triangle which are opposite the vertex H has been translated to; which seems
reasonable as we are translating H to overlap the original triangle.

Our
new circles all have equal radii and that distance is equal to the distance of
the Median of the new triangle ABC.

If
we label the points where the circles intersect which the center of the
circumcircles of triangles AHB and CHB, E and F and draw segments between the centers
of the circles a parallelogram is formed regardless of how we shift the
triangle ABC.