I understand that there are sets of 7 points on a circle that can be fully
shattered using triangles.But, it is not clear to me why it cannot shatter 8 points.


Is there any intuitive way of arriving at the conclusion that it can't shatter 8 points? Is there a simple explanation without using much advanced geometry tools?



Why the VC dimension of triangles in 2D space is not greater than 7?
Let $x\_1, \dots, x\_8$ be $8$ points on a circle in this order. Then there is no triangle containing exactly $x\_2, x\_4, x\_6, x\_8$ in its interior and the other four points in its exterior. Intuitively, it would have to intersect the circle in at least $8$ points, but any triangle intersects a circle in at most $6$ points.


