__Write-Up #8__:
Investigation of Altitudes and Orthocenters

If we construct any triangle ABC, and find
the orthocenter H, we can construct triangles with point H: BHC,
AHB, and CHA. (Click ** HERE**
for the script.)

It becomes obvious, if we construct the orthocenters
of each of these triangles (BHC, AHB, and CHA), that the __vertices
of the original triangle ABC__ are the orthocenters of the smaller
triangles.

The rationale for this occurrence is due to
construction of the orthocenter H. (See figure below.) Recall
that H was determined by the intersections of the projections
from Point A to the opposite side BC, from Point B to the opposite
side, and the same for Point C. Hence, __for triangle BHC__,
A is already the altitude of BHC. In order to construct the orthocenter
H, an altitude was constructed with Point C to side AB, so __Point
A__ is also perpendicular to the extension of HC. A similar
argument can be made for altitude from Point B to side AC so that
__Point A__ is also perpendicular to extension of BH. Hence,
Point A is the orthocenter of triangle BHC. (A similar argument
can be made for the other triangles involving Point H with respect
to the vertices of the original (red) triangle.)

If we construct circumcircles for triangles ABC, AHB, BHC,CHA and a Nine-Point Circle for ABC, then the resulting figure will appear as below. (Click HERE for the script.)

Note that the circumcircles make for an interesting figure. Of particular note are two types of symmetry for further exploration in this write-up: (1) symmetry about the legs of the original ABC triangle (red), and (2) symmetry about the legs of the orthotriangles AHB, BHC, and CHA (blue).

__Part 1. Symmetry About the Legs of the
Original ABC Triangle__

Note that the two circles through points AHC and ACB (see figure below) are reflected through line AC. This fact appears because the circumcircles C' and C'' were constructed through points A and C; hence, the circles C' and C'' have the same radii. The triangles ACC' and ACC'' are isosceles triangles with the same base. Hence the two circles have the same chord, the circles reflections of each other through the line AC, and the circles are congruent to each other. The same argument can be made about the other two sides of triangle ABC (red).

** Part 2. Symmetry About the Legs of Orthotriangles
AHB, BHC, CHA** Observe the three
pedals (black)

__Hence, we can conclude that all four
circumcircles are congruent to each other.__

** We can also conclude that Point A can
be interchanged with Point H **because,
as we mentioned before, the

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