Investigation by Samuel Obara

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.