It's nearly pancake day here in the UK and I'll be making my kids pancakes. The last one in the batch is always a little oddly shaped, and I'd like to divide it into two pieces of equal area so they can share it. However, the children are fussy eaters and insist on their pieces being path-connected. Is this possible? Maybe there's an old theorem somewhere that would help me have a stress-free day?
(n.b. I'm capable of making very precise and wiggly cuts with the tip of the knife, so model the cut as a continuous map $C:[0,1] \to \mathbb{R}^2$)