Multi-contrast brain MRI provide complementary soft-tissue characteristics that aid in the screening and diagnosis of diseases. However, limited scanning time, image corruption and various imaging protocols often result in incomplete multi-contrast images. While current approaches excel in image synthesis, they often struggle to synthesize critical tumor regions and exploit contextual information in multi-contrast brain MRI effectively. To address this issue, we propose a synthesis-centric, segmentation-assisted closed-loop framework with retrieval augmentation synthesis. Our method overall takes a generative adversarial architecture, which aims to synthesize missing contrasts from any combination of available ones with a single model. To explicitly capture tumor semantics and focus synthesis on tumor regions, we add an auxiliary segmentation branch that predicts tumor masks and feeds them back as semantic conditioning in synthesis branch, thereby learning tumor-aware representations in the model and improving synthesis fidelity. Furthermore, we propose a dual-bank retrieval augmentation strategy. It dynamically queries two external knowledge bases, namely a tumor masks memory bank for crucial tumor context and cross-image contrast feature memory bank for global style information, to augment synthesis. Verified on two public multi-contrast magnetic resonance brain datasets: BraTs2020 and UCSF-BMSR, the proposed method is effective in handling medical brain images synthesis tasks and shows superior performance compared to previous methods. Code is available at:https://github.com/iBizzard/SSCF.git