Final answer:
A great circle route is indeed the shortest distance between any two points on Earth's surface, which is true. This is because a great circle represents the maximum circle that can be drawn on a sphere, like Earth, differing from the straight line concept in the Pythagorean theorem applicable to flat surfaces.
Step-by-step explanation:
The statement that a great circle route is the shortest distance between any two points on Earth's surface is true. A great circle is any circle on the sphere's surface whose center coincides with the center of the sphere. Earth's equator is one example of a great circle, and others pass through both the North and South Poles, known as meridians, which are perpendicular to the equator and cross it at right angles.
When discussing travel across the Earth, especially in terms of air travel, following the arc of a great circle is indeed the shortest path one can take. The reason this is the shortest path is because Earth is a sphere and the great circle route represents the maximum circle that can be drawn on a sphere. For instance, a plane traveling from Chicago to Rome would follow this route as it is more efficient, covering a shorter distance than any other path on Earth's curved surface.
While it is common to think that the shortest distance between two points is a straight line based on the Pythagorean theorem, this principle applies to flat surfaces. On the surface of a sphere such as Earth, the shortest distance is the arc of a great circle, not a straight line between two points. This concept, originating from geometry, shows the importance of considering the actual shape of the Earth in navigation and travel.