In this paper, we have used two different proof techniques to show the Hamilton-connectedness of graphs. By using the vertex connectivity and Hamiltoniancity of graphs, we construct an infinite family of Hamilton-connected convex polytope line graphs whose underlying family of convex polytopes is not Hamilton-connected. By definition, we constructed two more infinite families of Hamilton-connected convex polytopes. As a by-product of our results, we compute exact values of the detour index of the families of Hamilton-connected convex polytopes. Finally, we classify the Platonic solids according to their Hamilton-connectedness and Hamilton-laceability properties.