Converse characterization of tileable ideal trapezoids

Prove that an ideal trapezoid ABCD is tileable by the integral-sided triangle (a,b,c), with pairwise coprime integers satisfying c^2=a^2+b^2+ab, if and only if its side parameters satisfy x>c^2−a−b and ab divides y.

Background

Proposition 2.3 establishes sufficiency of the conditions x>c2−a−b and ab|y for tiling an ideal trapezoid. The paper identifies proving the converse—necessity of these same conditions—as an intermediate step toward the broader equilateral conjecture.

References

Let $ABCD$ be an ideal trapezoid and $(a,b,c)$ be as required from our setup (that is, pairwise coprime integers with $c2 = a2 + b2 + ab$). Then $ABCD$ can be tiled by $(a,b,c)$ if and only if $x > c2-a-b$ and $(ab)|y$.

— Tiling Triangles with $2π/3$ Angles  (2512.22696 - Zhang, 27 Dec 2025) in Conjecture 4, Section 6, Conclusion and Future Work