Exploration 04  Tracing the Orthocenter
I performed a review of the locus of an orthocenter having a fixed side with the opposite side fixed to various line types which I will demonstrate and discuss in this writeup. Feel free to do your own exploration in my GSP file, which can be found here:
04_triangleexploration.gsp  
File Size:  16 kb 
File Type:  gsp 
First, the video above shows the first page of my gsp file, which is a simple construction of a triangle orthocenter in which no sides or points are fixed.
On the third tab, I traced the orthocenter with fixed line AB and point C moving along a line not parallel to AB. I am calling this line a linear trace. I found that the traced line of the orthocenter formed a rational function shape or hyperbola. It seems like these traces are making conic sections or shapes a degree higher than the lines point C is placed on.
On the next tab, I traced the orthocenter with fixed line AB and point C moving along a cubic function. I am calling this line a cubic trace. I found that the traced line of the orthocenter formed a shape of what appears a rational function or hyperbola type of shape. Perhaps these traces are making conic sections. This might make sense since a triangle is similar to the shape of a cone.

On the next tab, I traced the orthocenter with fixed line AB and point C moving along a line parallel to AB. I am calling this line a constant trace. I found that the traced line of the orthocenter formed a parabola.
On the next tab, I traced the orthocenter with fixed line AB and point C moving along a parabola. I am calling this line a quadratic trace. I found that the traced line of the orthocenter formed a shape of what appears to be a cubic function, an inverse cubic function to be exact. This does not appear to be a conic section, but I might need to trace further to ensure that it is not.
