Vanishing Point Theorem, short version:
If two or more lines in the real world are parallel to each other, but not parallel to the picture plane, then they have the same vanishing point. That is, the perspective images of these lines will not be parallel, and if fully extended, the image lines will intersect. The intersection point is the vanishing point.
In this drawing, we see a road, with lots of wagon wheel tracks. From our experience in real life, we feel comfortable assuming that the sides of the road are parallel to each other, and based on the drawing, we also feel comfortable assuming the wagon wheel tracks are not only parallel to each other, but also to the sides of the road.
From the Vanishing Point Theorem, we know that if we extend these lines until they intersect, and if the painting is drawn in rigorous perspective, we should find that all of them intersect at the same point. That point will be the vanishing point.
