Our investigation begins with the construction of the pedal triangle. The first step is to create the triangle ABC and pick an arbitrary point in the same plane.
The next step is to extend the sides of the triangle and connect them to the pedal point by 'dropping' a perpendicular line.
The pedal triangle is created by the feet of the altitude, in other words, the point of intersection of the lines connecting the pedal point and the extended sides of the triangle. The following image show the point of intersection without the pedal triangle. Points R,S, and T are the points of intersection. They will be the vertices of our pedal triangle.
The exploration that I have chosen focuses on the locus of the midpoints of the pedal triangle as the pedal point travels along the path of an excircle of a triangle.
We need first to create the excircle of our triangle ABC. The term excircle suggests that our circle will be on the exterior (or outside) of the triangle. This, in fact, is true. In addtition, the excircle has the property that it is tangent to all of the sides of the triangle. In two cases the point of tangency happens on the extension of the sides of the triangle.
Begin with any triangle ABC (with extended sides) and create the angle bisectors for each vertex.
The center of the excircle is the intersection of the exterior bisector of <A and <C and the interior bisector of <B. For this intersection, we need to create a perpendicular line to both angle bisectors of <A and <C.
The construction of the excircle is completed by creating a perpendicular from the intersection to one of the extended sides. This becomes the radius of the excircle. You can click on the image below to see that the excircle is tangent to each of the sides for any triangle.
We need to combine the two concepts that we have constructed in the beginning of our exploration. The investigation calls for the observation of the pedal triangle as the pedal point travels the path of the excircle.
More specifically, we will be constructing the locus of the midpoints of the sides of the pedal triangle. The image below shows the locus. You can click on the image to see the creation of the locus.
It is interesting to note that the angle bisectors (internal and external) become one of the axes for the ellipse. In two of the cases, the bisectors are the major axes and in one case it is the major axes. In furthur exploration of various triangles, I found that this general case holds, that is two bisectors are one type of axes and the one bisector is the other type of axes.
After noticing the relationship between the angle bisectors and the axes of the ellipse, I wanted to explore the location of the foci for each of the ellipse. I started first with the equilateral triangle as a special case and added in some lines previously hidden. More specifically, I focused my attention on the points of tangency and intersection of the excircle and the triangle ABC.
The image above suggests that the foci of the ellipses are the points of tangency on the excircle and the sides of the triangle. The foci of the middle ellipse are the point of tangency and the intersection of the bisectors. I then changed my triangle to see if this theory held true for any arbitrary triangle.
As I explored the various triangles and the affect it had on the foci of the ellipse, I was not able to determine a specific pattern.It did seem that the points of tangency and points of intersection became the foci for many of the ellipse, but not all and not consistently. I suggest that you explore the various triangles for yourself. If you wish to do so, you may click on the above image.