On Positive Integer Descartes-Steiner Curvature Quintuplets

In Descartes' five circle problem integer curvatures (inverse radii) are considered. The positive integer curvature triple [c_1, c_2, c_3] (dimensionless), with non-decreasing entries for three given mutually touching circles, leading to integer curvatures [c_{4,-}, c_{4,+}] for the two circles touching the given ones is called a Descartes-Steiner triple. They come in two types: [c, c, d] (or [c, ,d, d]) and triples with distinct entries. The first case is related to Pythagorean triples. The distinct curvature case is more involved and needs a combined representations of certain binary quadratic forms of the indefinite and definite type. The degenerate case when a straight line touches the three given touching circles can also be characterized completely.