The guys over at Numberphile have a great video on the Euler line of a triangle. At the time referenced at the following link, they use a computer simulation to show the movements of the orthocenter, medicenter, and circumcenter of a triangle as it changes shape. As I watched the video, the movements of the triangle and its Euler line started to look three dimensional.
https://youtu.be/wVH4MS6v23U?t=417
I have seen a proof of of the statement "every triangle is the orthogonal projection of some equilateral triangle." However, I cannot find a reference to Euler lines of the projection and whether there is a corresponding line through the equilateral triangle that correlates to the Euler line of the projection.
So the full question is thus: Can a triangle and its Euler line be described as a projection into two dimensions of an equilateral triangle and an orthogonal line through its medicenter sitting in three dimensions? Or is my intuition just completely off?