Imagine a simple X / Y coordinate graph. A circle surrounds the point of origin. Let's say the radius = 3.
We want to know how many points exist on the circumference of the circle through which a line may be drawn from the point and cross the origin.
Here, we simply find the circumference of the circle, which is 18.85
Simple enough. But now, what if we draw a square within this circle? The perimeter of this inscribed square would be 16.97, which is smaller than the circumference of the outside circle.
But: if the perimeter of this inscribed square is smaller and therefore fewer points exist through which a line may be drawn through + point of origin, how do we reconcile this with the fact that every point which may be drawn on the outlying circle's larger circumference MUST ALSO cross through the square's perimeter?
For every point on the circumference of the circle through which a line may be drawn through the point of origin, a point must exist on that line which also lies on the perimeter the inner square. Therefore, the same amount of points must exist on the perimeter of the square.