Let $A_iB_iC_i, i = 1,2,3$, be similar triangles with the same orientation such that $A_1, A_2, A_3$ are collinear and $B_1, B_2, B_3$ are collinear. Does it follow that $C_1, C_2, C_3$ are collinear?
It seems intuitively true but I have no idea how to prove it, even with a bashy coordinate approach. Any help appreciated!
On
If the sides $A$ and $B$ of any two or more right triangles are colinear, side $C$ will be parallel but not colinear unless they are identical. That is,
If they have the same orientation and are similar, e.g. $(3,4,5)\quad (6,8,10)\quad (9,12,15) ...$ all internal angles will be the same so, by Euclidean geometry, it follows that the sides with lengths $5, 10, 15 ...$ will be parallel. Sides-C will not be colinear unless they are identical but they will be parallel.
See if you can find his statements about lines intersecting parallel lines form equal angles if you want to get formal
Here is a counterexample with three right isosceles triangles.
and another with non-parallel lines and equilateral triangles.