Gamble's Theorem

Gamble’s Theorem states that for any closed shape in a plane, the maximum or minimum length of a curve passing through a specific set of points (with fixed endpoints) is determined by the arrangement of these points and the shape's boundaries. Essentially, it connects the shortest or longest possible path connecting certain points within a region, emphasizing how geometric constraints influence the optimal path. This theorem helps in understanding how paths can be optimized within given boundaries, with applications in fields like geometry, physics, and engineering.