Question 3a
By Krista Floer
Does the path create a pattern of similar triangles?
The path does create a pattern of similar triangles. This happens when the starting point is anywhere but a midpoint or a trisection point. We have already shown that certain triangles are ALWAYS congruent, but we are looking at different triangles for this proof.

Something interesting to note: The path is the same if the starting point is placed between B and the midpoint as the midpoint and C. For the paths to be the same, not just any point can be chosen. The point must be reflected over the perpendicular line to BC that goes through the midpoint. Furthermore, the second point on each side from Barney's path no matter where the starting point is, follows the previous guideline for the reflection.
We can see that a, if reflected over the perpendicular line to side AB, lies on point d.