You end up getting solutions like this when two sides of your light triangle are the same distance, and then it will always simplify to 0 = 0. Two sides of a right triangle cannot be the same size and it still remain a right triangle. Because then they would have to intersect the third side at the same point, so then it would just be a line and not a triangle. That is one reason why it is important to assign different variables to different frames in relativity that cannot be exchanged for each other.
It doesn't reduce to 0=0. Clearly you've never done any Lorentz transform calculations, which means you'll be unable to do any of the questions asked in that problem sheet I just linked to.
You've obviously failed to understand what is going on here, a result of the problem you have with thinking in terms of spatial lengths, that's a very Newtonian way to view things. You can get away with it somewhat with the light clock example but it doesn't work in more general cases. The Euclidean metric is $$ds^{2} = dx^{2} + dy^{2} + dz^{2}$$, with time not coming into it. Any Galilean transform will preserve this metric, ie translations and rotations. Special relativity modifies this to say "It is not the spatial part which is invariant and the time part separately invariant but they form a single invariant combination $$ds^{2} = -dt^{2} + dx^{2} + dy^{2} + dz^{2}$$". Now you have to consider not just the length of a path in space but also in time. Obviously any transform which leads the time part alone and leaves the spatial part invariant will leave that combination invariant, ie translations and rotations, but there are now additional transformations, namely boosts, which leave the combination invariant by mixing the time and space parts.
Anyone who has done any Euclidean geometry will know how Pythagoras's theorem generalised to N dimensions, ie if you move $$dx$$ in one direction, $$dy$$ in another and $$dz$$ in the third then you've moved $$ds$$ in total from your starting place where $$ds^{2} = dx^{2} + dy^{2} + dz^{2}$$. This is just the Euclidean distance. Someone in a different set of coordinates, linked to yours by a Galilean transform, will see you to have moved by an amount dx' in the x' direction, dy' in the y' direction and dz' in the z' direction but and while $$dx \neq dx'$$ etc might be the case we know that $$dx^{2} + dy^{2} + dz^{2} = (dx')^{2} + (dy')^{2} + (dz')^{2}$$ by construction of the Galilean transforms. All the sides of this 'cube' formed from the 3 directions might change but the diagonal has the same length.
Fine for Euclidean geometry but in special relativity you also consider the amount of time it took to do that, dt. Special relativity says the length you have moved in space-time from your initial location is $$ds^{2} = -dt^{2} + dx^{2} + dy^{2} + dz^{2}$$. This is a single length, a single little line element. Doing a Lorentz transform then gives different dt', dx', dy', dz' but the diagonal length in this non-Euclidean space is unchanged. If you don't like thinking about non-Euclidean structures (or more likely cannot) then consider that $$-dt^{2} + dx^{2} + dy^{2} + dz^{2} = -(dt')^{2} + (dx')^{2} + (dy')^{2} + (dz')^{2}$$ can rearrange to $$(dt')^{2} + dx^{2} + dy^{2} + dz^{2} = dt^{2} + (dx')^{2} + (dy')^{2} + (dz')^{2}$$. This we can now view in the same manner as the Euclidean case, we have 2 4d cubes with side lengths dt',dx,dy,dz and dt,dx',dy',dz' and although $$dt' \neq dt$$ etc might be the case the length of the diagonal from one corner to another is the same for each cube.
That is a highly non-trivial property, not this "oh it becomes 0=0" nonsense you said. Lorentz transforms modify and warp the cubes but
always preserve the diagonal in the construction I've just given. If you have a transform $$(dt,dx,dy,dz) \to (dt',dx',dy',dz')$$ which results in the two cubes $$(dt',dx,dy,dz)$$ and $$(dt,dx',dy',dz')$$
not having the same diagonal length then you haven't got a Lorentz transform. The fact we've had to swap dt and dt' to give it a Euclidean interpretation (so
you can understand it) makes this somewhat inelegant but clearly grasping non-Euclidean geometry is something you haven't gotten around to yet (I doubt you're capable of it anyway). The use of the geometric models of Minkowski, which you lambasted, makes all of this straight forward. You can view the Lorentz transforms as linear maps on a matrix of metric components, $$\eta_{ab} \to \Lambda^{c}_{a}\eta_{cd}\Lambda^{d}_{b} = \eta_{ab}$$. If the matrix isn't left unchanged then you don't have a Lorentz transform. That way you can avoid having to think about lengths of lines in non-Euclidean space and just crunch through some matrix-matrix products. But then something tells me you cannot do any matrix mathematics either.....