Summary: This paper investigates the output-based event-triggered communication and control for linear multi-agent consensus under a directed graph based on a co-design method. The communication among the agents, as well as the controller updates, are determined by some new event-triggering mechanisms, in order to reduce the use of the network resources. To simultaneously guarantee the significant properties including the asymptotic consensus, and strong Zeno-freeness (strictly positive inter-event times), a novel distributed dynamic event-triggered protocol is proposed. Unlike the most-existing emulation-based approaches in which the control gain is previously decided, a systematic co-design procedure is proposed to design the controller gain, the observer gain, and the event-triggering mechanisms altogether in terms of solving a linear matrix inequality optimization problem. Based on the resulting hybrid system framework, a hybrid model is established for the distributed closed-loop system and the asymptotic consensus is achieved. Finally, a numerical example is presented to verify our systematic design methodology.