This article addresses the resilient consensus problem of multiagent systems subject to cyber attacks on communication links, where the attacks on different links may collude to maintain undetectable. For the case with noncollusive attacks on links, a distributed fixed-time observer is designed so that the attack on each link can be detected by the two associated agents. A necessary and sufficient condition is derived to ensure the isolation of attacked links and no mistaken isolation of normal ones. For the case with collusive attacks on links, a novel attack isolation algorithm is proposed by constructing extra observers on the basis of the previous designed distributed fixed-time observer via sequentially removing the information associated with one of the links. Based on the isolation of the attacked links, a control algorithm is designed, and a necessary and sufficient condition is provided to achieve resilient consensus. Numerical examples corroborate the effectiveness of the proposed strategies.