Final answer:
Hamilton path and Hamilton circuit are graph theory terms where both visit each vertex once, but a circuit also returns to the starting point. The provided context, however, references Alexander Hamilton and race track strategies, which do not directly relate to these mathematical concepts.
Step-by-step explanation:
The terms Hamilton path and Hamilton circuit are concepts from the field of graph theory in mathematics. A Hamilton path is a path in an undirected or directed graph that visits each vertex exactly once. A Hamilton circuit, also known as a Hamiltonian cycle, is a Hamilton path that is a cycle, meaning it returns to the starting vertex.
In the context provided, it seems there is a mix-up with the historical figure Alexander Hamilton and the concept of Hamilton paths and circuits. The portion regarding racing and cutting corners is more related to physical principles like speed and distance, possibly indicating a physics problem about motion on a race track. However, as there's no specific graph theory problem given, it's challenging to directly apply the concepts of Hamilton paths or circuits to the race track scenario without additional context.