**This investigation goes around the altitude
of a triangle. The altitude of any given triangle will meet at
a given point (orthocenter 'H'). To see a GSP sketch that demonstrate
this idea click here. The point
where the altitude meet is called the orthocenter of the triangle.
We can obtain three triangles by connecting H to the vertices
of triangle ABC as shown below.**

**It can be noted that the orthocenter of
HAC is at point B, click
here to see this fact. For triangle HAB, it has its orthocenter
at point C, as it can be illustrated by clicking here.
Finally the orthocenter of HBC is at point A also click here to see for yourself.
It is interesting to note that the orthocenters of any of the
orthic triangles falls on the vertex of the original trianglre
that is not a point in the orthic triangle in question. One interesting
point I have noted is that when we draw the circumcircles of each
of the four triangles, the circles appear to be the same. See
below for a diagram or click here
to access the GSP file used to create the image.**

**Since the circles appeared to be congruent,
I used GSP to find this areas. The areas of the four circles were
all equivalent, I started to try to find a proof of this fact.
To start with, I decided to consider two of the circumcircles.
Let us use the circle about triangle ABC and the circle about
triangle HAC. Proving these two congruent will lead to the fact
that all four circumcircles must be congruent, as the reasoning
applied to the circle about triagle HAC would easily fit for the
circles about the other orthic triangles.**

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