Wetzel's 30-60-90 Triangle Covers Unit Arcs

Abstract

John E. Wetzel conjectured that the 30-60-90 triangle T obtained by placing a square of side 1/3 on the hypotenuse covers every unit arc in the plane. We give a computer-assisted proof of this conjecture with independently checkable interval certificates. The proof reduces a hypothetical noncovered arc to a finite family of 599 closed second-order cone models, covering all representative and raw tail-order branches, and certifies a polygonal-chain lower bound greater than one in every model by interval validation of stored dual certificates. Since every certified lower endpoint exceeds 1.0048, the homothetic copy T/1.0048 still covers every unit arc. Its area is 0.260956..., below the area pi/12 approx. 0.261799 of the 30-degree unit sector, a certified area improvement over the sector cover within this convex Wetzel-cover setting.