Geometric arithmetic: triangular number triples

Question

Call a triple $x, y,$ and $z$ of numbers triangular if and only if there is a triangle whose sides are in the triple ratio $x:y:z$. Since the sum of two sides of a triangle exceeds the remaining side, it is clear that if triple $x, y,$ and $z$ of numbers is triangular, then $x + y > z$ , $y + z > x,$ and $z + x > y$ . Is the converse true?

Answer 1

Yes. Without loss of generality, suppose that $z$ is the longest of the three numbers. Draw a line segment $AB$ having length $z$. Now draw circles with radii $x$ and $y$ centred at $A$ and $B$ respectively. These circles will intersect at points $P$ and $Q$. The triangles $PAB$ and $PAQ$ have side lengths $x,y,z$.