Measurements of the Branching Fractions of B0 --> K^*0K+K-, B0 --> K^*0pi+K-, B0 --> K*0K+pi-, and B0 --> K*0pi+pi-

Branching fraction and asymmetry measurements of charmless $B^0\to K^{*0}h^+_1h^-_2$ ($h_{1,2}$ = $K$, $\pi$) decays are presented, using a data sample of 383 million $\Upsilon(4S) \to$ $B\bar{B$} decays collected with the BaBar detector at the PEP-II asymmetric-energy $B$-meson factory at SLAC. The results are: ${\cal B}$($B^0 \to K^{*0}K^+ K^-)$ = (27.5 $\pm$ 1.3 $\pm$ 2.2) $\times$ 10$^{-6}$, ${\cal B}$($B^0$ $\to$ $K^{*0}\pi^+ K^-$) = (4.6 $\pm$ 1.1 $\pm$ 0.8) $\times$ 10$^{-6}$ and ${\cal B}$($B^0$ $\to$ $K^{*0}\pi^+\pi^-$) = (54.5 $\pm$ 2.9 $\pm$ 4.3) $\times$ 10$^{-6}$. The first errors quoted are statistical and the second are systematic. An upper limit is set for ${\cal B}$($B^0$ $\to$ $K^{*0}K^+ \pi^-$) $<$ 2.2 $\times$ 10$^{-6}$ at 90% confidence level. We also present measurements of CP-violating asymmetries for the observed decays.