A Zeno line is a straight line in a fluid's temperature-density plane where the compressibility factor equals unity and stretches from supercritical to subcritical regimes. A second unexpected linearity also occurs in liquids in their normal liquid range (NLR) where the compressibility factor approaches zero. This Zeno-like line includes liquids as diverse as monatomic and diatomic elements to hydrogen bonding liquids, with water and helium being the only currently known exceptions. Remarkably, this second linearity is also present in polymeric liquids, molten metals, and salts, i.e., it is a generic characteristic of liquids with negligible or low vapor pressures. This observation yields the following new corresponding states principle: Saturated liquid densities are a linear function of temperature in the NLR for molecular, polymeric, inorganic, ionic, and metallic liquids and superpose to form a single master curve. Another Zeno-like linearity in the NLR has also been identified for the configurational energy of many liquids. Extension of this line to zero temperature defines the ground state configurational energy of the hypothetical disordered liquid (glass).