I'm reading this explanation of why the pedal triangle has the smallest perimeter of any triangle you can inscribe in an acute triangle. The figure in that linked page (Fig. 21) shows a pedal triangle $EFG$ and some other triangle $UVW$ both inscribed in a bigger, acute triangle $ABC$. The triangle $ABC$ is then reflected six times as shown, and the inscribed triangles are reflected along with it.
My question is: Why can't there be another, non-pedal triangle whose perimeter (well, two times the perimeter) also traces out a straight line when we reflect it?
(Sorry if this is a dumb question, I'm just reading this book for fun and couldn't figure out why we needed the sides to be the altitudes for the sides to form a straight line when reflection.)