Construction and uniqueness
Side a stays fixed while B and C move along the perpendicular
rays at its endpoints. The sides b and c start perpendicular
to B and C and point inward.
With the checkbox selected, b and c keep their prescribed lengths.
The tentative side A joins their free endpoints. Four right angles
are automatic; the displayed two angles must also become 90°.
Away from the solution, the trial polygon can cross itself.
With the checkbox cleared, A is constructed as the common
perpendicular of the supporting geodesics of b and c, whenever
this gives a convex right hexagon. The resulting lengths b and c
are displayed.
Any right hexagon with the prescribed a, b, c must satisfy:
cosh B = (cosh b + cosh a cosh c) / (sinh a sinh c)
cosh C = (cosh c + cosh a cosh b) / (sinh a sinh b).
Each right-hand side is greater than 1, so these equations
specify exactly one positive B and C. At these settings the
construction closes with all six angles right. Thus the
prescribed lengths determine a hexagon uniquely up to isometry.
Lengths use curvature −1. The disk model preserves angles;
distances near its boundary are visually compressed.