Large shared displays, such as digital whiteboards, are useful for supporting co-located team collaborations by helping members perform cognitive tasks such as brainstorming, organizing ideas, and making comparisons. While recent advancement in Large Language Models (LLMs) has catalyzed AI support for these displays, most existing systems either only offer limited capabilities or diminish human control, neglecting the potential benefits of natural group dynamics. Our formative study identified cognitive challenges teams encounter, such as diverse ideation, knowledge sharing, mutual awareness, idea organization, and synchronization of live discussions with the external workspace. In response, we introduce LADICA, a large shared display interface that helps collaborative teams brainstorm, organize, and analyze ideas through multiple analytical lenses, while fostering mutual awareness of ideas and concepts. Furthermore, LADICA facilitates the real-time extraction of key information from verbal discussions and identifies relevant entities. A lab study confirmed LADICA's usability and usefulness.