$3n$ points are drawn on a flat piece of paper, such that no $3$ points lie on a straight line. Is it always possible to connect triples of points with straight lines, such that you form $n$ triangles and no two triangles intersect or touch each other?
Is it possible?
Yes it is always possible.
Order your points by x-coordinate (and by y for those with equal x - if any).
Take three points at a time from the ordered list to form each triangle.
The first triangle obviously won't intersect any other.
For each subsequent triple, its points are entirely to the right of all triangles created so far (or at least as far right and above - for the equal x case), so edges between them are guaranteed to not intersect existing triangles.