The generation of high-quality entangled photon pairs has been a long-sought goal in modern quantum communication and computation. So far, the most widely used entangled photon pairs have been generated from spontaneous parametric down-conversion (SPDC), a process that is intrinsically probabilistic and thus relegated to a regime of low rates of pair generation. In contrast, semiconductor quantum dots can generate triggered entangled photon pairs through a cascaded radiative decay process and do not suffer from any fundamental trade-off between source brightness and multi-pair generation. However, a source featuring simultaneously high photon extraction efficiency, high degree of entanglement fidelity and photon indistinguishability has been lacking. Here, we present an entangled photon pair source with high brightness and indistinguishability by deterministically embedding GaAs quantum dots in broadband photonic nanostructures that enable Purcell-enhanced emission. Our source produces entangled photon pairs with a pair collection probability of up to 0.65(4) (single-photon extraction efficiency of 0.85(3)), entanglement fidelity of 0.88(2), and indistinguishabilities of 0.901(3) and 0.903(3) (brackets indicate uncertainty on last digit). This immediately creates opportunities for advancing quantum photonic technologies.