Painter Problem

Chef is a painter. He likes to draw geometric figures and among them, he especially loves isosceles right triangles. The longest side of a right triangle is called its hypotenuse. Chef drew $N$ isosceles right triangles (numbered $1$ through $N$) on an infinite sheet of paper, which can be represented by a Cartesian half-plane (containing all points $(x, y)$ such that $y \ge 0$), in such a way tha