Though, still for the 45-45-90 I don't think you can pick a different angle? At least for alpha > 45 (because it also doesn't work for this, the lines diverge), you can always swap it so beta is > 45 for those cases. If you pick something other than 90 to be 2 alpha, the reflection mentioned in step 1 can't be done
Not sure how much that helps with "infinite triangle" with two 90deg and one 0deg angles.
That's btw how you get sin90=1 which doesn't have any geometrical sense when we consider finite triangles.
Or in case of triangle with 45, 45, 90 maybe you could just pick different angle than 90 to be 2alpha.