  We present the results of a large survey of HI, OVI, and CIII absorption
lines in the low-redshift (z<0.3) intergalactic medium (IGM). We begin with 171
strong Lyalpha absorption lines (W>80 mA) in 31 AGN sight lines studied with
the Hubble Space Telescope and measure corresponding absorption from
higher-order Lyman lines with FUSE. Higher-order Lyman lines are used to
determine N_HI and b_HI accurately through a curve-of-growth (COG) analysis. We
find that the number of HI absorbers per column density bin is a power-law
distribution, dN/dN_HI=N^-beta, with beta_HI=1.68+-0.11. We made 40 detections
of OVI 1032,1038 and 30 detections of CIII 977 out of 129 and 148 potential
absorbers, respectively. The column density distribution of CIII absorbers has
beta_CIII=1.68+-0.04, similar to beta_HI but not as steep as beta_OVI=2.1+-0.1.
  From the absorption-line frequency, dN_CIII/dz=12^+3_-2 for W(CIII)>30 mA, we
calculate a typical IGM absorber size r_0~400 kpc. The COG-derived b-values
show that HI samples material with T<10^5 K, incompatible with a hot IGM phase.
By calculating a grid of CLOUDY models of IGM absorbers with a range of
collisional and photoionization parameters, we find it difficult to
simultaneously account for the OVI and CIII observations with a single phase.
The observations require a multiphase IGM in which HI and CIII arise in
photoionized regions, while OVI is produced primarily through shocks. From the
multiphase ratio N_HI/N_CIII, we infer the IGM metallicity Z_C=0.12 Z_sun,
similar to our previous estimate of Z_O=0.09 Z_sun from OVI.
