Why Delta Complex structure?

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I am a bit confused about delta complex structure and CW complex structrue.
I was wondering is n-dimensional disks homeomorphic two n dimensional triangles?
I mean if that is the case then I can make every Delta complex structure to a CW complex structure and vice versa. If this is the case one could ask why do we have the delta complex structure? Could the hole homology theory been built from CW complexes? or lets say with some other complex structure i.e from n dimensional squares?