Why eukaryotes have three different types of cytoskeleton filaments?
While eukaryotes contain tubulin-containing microtubles and actin filaments, prokaryotic homologues do exist. 
For example FitZ protein, the most common bacterial Tubulin homologue[1], is involved in cell division. Genes for this protein are actually present in eukaryotic nuclear DNA and the protein facilitate the division of the mitrochrondria and chloroplast organelles which were created by endosymbiosis[2]. 

Similarly, MreB, functions as the most common prokaryotic analogue of actin.

Regarding IF filaments, certainly subsets such as nuclear lamins are uniquely found within Eukarya, however a paper published by Bafchi[4] has suggested that IF-like proteins are present within bacteria. 


Bi, E., &amp; Lutkenhaus, J. (1991). FtsZ ring structure associated with division in Escherichia coli. Nature, 354(6349), 161-164. doi:10.1038/354161a0
Margolin W. FTSZ AND THE DIVISION OF PROKARYOTIC CELLS AND ORGANELLES. Nature reviews Molecular cell biology. 2005;6(11):862-871. doi:10.1038/nrm1745.
Gunning, P. W., Ghoshdastider, U., Whitaker, S., Popp, D., &amp; Robinson, R. C. (2015). The evolution of compositionally and functionally distinct actin filaments. Journal of Cell Science, 128(11), 2009-2019. doi:10.1242/jcs.165563
Bagchi S, Tomenius H, Belova LM, Ausmees N. Intermediate filament-like proteins in bacteria and a cytoskeletal function in Streptomyces. Molecular Microbiology. 2008;70(4):1037-1050. doi:10.1111/j.1365-2958.2008.06473.x.