***Before reading this write up, it will be helpful to open this GSP file!!***
In the linked GSP file, you will see the yellow triangle RST is what we call our Pedal Triangle. I have created midpoints on each of the three sides of this triangle. These will later be traced. The next thing the question asks us to do is to construct a circle with the center being the circumcenter of triangle ABC. To create the circumcenter of triangle ABC, I will create perpendicular bisectors of the triangle and the intersection of these three lines is the circumcenter, which I have labeled as point I. Now with point I as our center, we will create a circle whose radius is larger than the radius of the circumcircle. The circumcircle of triangle ABC is the circle seen in purple and so is its radius.
Now using the circumcenter, point I, we want to construct another circle whose radius is larger than the radius. So I will just plot a random point, named Z, outside of the purple circumcircle and create a segment IZ to be our radius. Our bigger circle and radius is seen in orange.
Finally, I have set our midpoints to have traces and merged point P onto our orange circle and animated it around this circle. In other words, we now have our pedal point animated around our circle and the midpoints of our Pedal Triangle are being traced. Press play on the motion controller to see that the three paths created by the traces of the midpoints are 3 ellipses. (Note: make sure when tracing the midpoints to see the three paths, only point P is selected before pressing play on the motion controller).
Hopefully once you've traced the points, your GSP file you've opened looks like this, where the three red ellipses are the 3 paths we were looking for:
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