Mathematics

The key insight is that distance relationships from a point to endpoints of a segment are connected through the midpoint. Since $C$ divides $AB$ equally, you can use the triangle inequality and the geometry of how $P$ relates to both $A$ and $B$ simultaneously. Think about expressing $PA$ and $PB$ in terms of $PC$ and the known segment $AC = CB$—this often reveals a constraint or identity that holds for any position of $P$.

The key insight is that distance relationships from a point to endpoints of a segment are connected through the midpoint. Since $C$ divides $AB$ equally, you can use the triangle inequality and the geometry of how $P$ relates to both $A$ and $B$ simultaneously. Think about expressing $PA$ and $PB$ in terms of $PC$ and the known segment $AC = CB$—this often reveals a constraint or identity that holds for any position of $P$.