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We present the results of a large survey of H I, O VI, and C III absorption lines in the low-redshift (z<0.3) intergalactic medium (IGM). We begin with 171 strong Lyα absorption lines (W_λ≥80 mÅ) 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 H I absorbers per column density bin is a power-law distribution, dN/dN_HI_∝N-β^_HI_, with β_HI_=1.68±0.11. We made 40 detections of O VI λλ1032, 1038 and 30 detections of C III λ977 out of 129 and 148 potential absorbers, respectively. The column density distribution of C III absorbers has β_CIII_=1.68±0.04, similar to β_HI_but not as steep as β_OVI_=2.2±0.1. From the absorption-line frequency, dN_CIII_/dz=12+3^_–2for W_λ(C III)>30 mÅ, we calculate a typical IGM absorber size r₀∼400 kpc, similar to scales derived by other means. The COG-derived b-values show that H I samples material with T<10⁵ 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 O VI and C III observations with a single phase. Instead, the observations require a multiphase IGM in which H I and C III arise in photoionized regions, while O VI is produced primarily through shocks. From the multiphase ratio N_HI_/NCIII_, we infer the IGM metallicity to be Z_C=0.12 Z_☉, similar to our previous estimate of Z_O=0.09 Z_☉from O VI.