Triangles constructed from tangent circles: auxiliary, centre, and contact

This expository note organizes, around a single question, the classical metric identities associated with configurations of mutually tangent circles. The question is: which triangles can be constructed from tangent circles, and how do the radii determine their shape? Three constructions are studied. Two circles with a common external tangent produce the auxiliary triangle, with sides r₁+r₂, r₁−r₂ and 2√(r₁r₂), and yield a bijection between externally tangent configurations and marked right triangles. Three mutually tangent circles produce the centre triangle, whose sides are the pairwise sums of the radii, and establish a bijection with all triangles (Ravi substitution). Their points of tangency form the contact triangle, which is always acute and establishes a bijection, up to similarity, with acute triangles; its angles satisfy rᵢ = ρ·tan(αᵢ), with ρ its circumradius. It is shown that the common tangent line may be regarded as a circle of zero curvature: the triangle GTN is right-angled at T, with hypotenuse 2√(r₁r₂) and angle at G equal to half the angle of the auxiliary triangle at O₁, and appears as the limit of the contact triangle as the third radius tends to infinity. Descartes' theorem with zero curvature closes the connection. Two exact figures and one schematic are included.