Medical image segmentation is fundamental to clinical decision-making, yet existing models remain fragmented. They are usually trained on single knowledge sources and specific to individual tasks, modalities, or organs. This fragmentation contrasts sharply with clinical practice, where experts seamlessly integrate diverse knowledge: anatomical priors from training, exemplar-based reasoning from reference cases, and iterative refinement through real-time interaction. We present $\textbf{K-Prism}$, a unified segmentation framework that mirrors this clinical flexibility by systematically integrating three knowledge paradigms: (i) $\textit{semantic priors}$ learned from annotated datasets, (ii) $\textit{in-context knowledge}$ from few-shot reference examples, and (iii) $\textit{interactive feedback}$ from user inputs like clicks or scribbles. Our key insight is that these heterogeneous knowledge sources can be encoded into a dual-prompt representation: 1-D sparse prompts defining $\textit{what}$ to segment and 2-D dense prompts indicating $\textit{where}$ to attend, which are then dynamically routed through a Mixture-of-Experts (MoE) decoder. This design enables flexible switching between paradigms and joint training across diverse tasks without architectural modifications. Comprehensive experiments on 18 public datasets spanning diverse modalities (CT, MRI, X-ray, pathology, ultrasound, etc.) demonstrate that K-Prism achieves state-of-the-art performance across semantic, in-context, and interactive segmentation settings. Code will be released upon publication.