Invariants and areas of Steiner 4-chains

The paper rigorously proves that, among poristic Steiner 4‑chains with fixed Soddy circles, the area is maximized at the axial symmetric chain C** and minimized at the two lateral symmetric chains, providing explicit closed forms for the extremal areas; see Theorem 2 and its proof outline, which differentiates S = Σ r_i^2 with respect to one curvature t and shows the only critical points occur at the three symmetric configurations, then identifies max/min among them . The candidate solution obtains the same formulas and the same extremal locations but uses a different method: it fixes I(4)_1, I(4)_2, I(4)_3 (whose explicit expressions are given in the paper) and expresses S as a function of σ4 = Π b_i, then invokes Newton–Maclaurin to prove S is strictly decreasing in σ4, and finally evaluates σ4 at the axial and lateral chains (whose curvatures the paper also computes) to conclude the ordering and the extremal values . The model’s proof is essentially correct but has a minor gap: it does not explicitly argue that σ4 attains its global minimum/maximum exactly at those symmetric chains; the paper fills this by a critical‑point analysis showing no other extrema exist.