The structural and dynamical properties of lipid membranes rich in phospholipids and cholesterol are known to be strongly affected by the unsaturation of lipid acyl chains. We show that not only unsaturation but also the position of a double bond has a pronounced effect on membrane properties. We consider how cholesterol interacts with phosphatidylcholines comprising two 18-carbon long monounsaturated acyl chains, where the position of the double bond is varied systematically along the acyl chains. Atomistic molecular dynamics simulations indicate that when the double bond is not in contact with the cholesterol ring, and especially with the C18 group on its rough beta-side, the membrane properties are closest to those of the saturated bilayer. However, any interaction between the double bond and the ring promotes membrane disorder and fluidity. Maximal disorder is found when the double bond is located in the middle of a lipid acyl chain, the case most commonly found in monounsaturated acyl chains of phospholipids. The results suggest a cholesterol-mediated lipid selection mechanism in eukaryotic cell membranes. With saturated lipids, cholesterol promotes the formation of highly ordered raft-like membrane domains, whereas domains rich in unsaturated lipids with a double bond in the middle remain highly fluid despite the presence of cholesterol.