The Variance–Deficit Exchange Identity for the Thales Right Triangle

This note derives an exact geometric identity for right triangles inscribed in a semicircle (Thales configuration), relating the lateral displacement of the apex to its height deficit from the symmetric configuration. The identity L^2 = D(2R - D) follows directly from the Pythagorean constraint and expresses a precise trade-off between asymmetry and optimal height. Interpreting the squared lateral displacement as a variance-like quantity and the height shortfall as a symmetry deficit, the identity reveals a quadratic cost structure governing deviations from symmetry. Near the symmetric configuration, small lateral deviations incur only second-order penalties, while larger deviations are increasingly constrained. Normalized, scale-free forms of the identity enable comparison across systems of different sizes and motivate a diagnostic classification into symmetric, operating, and degenerate regimes. The result is purely geometric and static: it introduces no dynamics, universality claims, or physical laws. Its contribution lies in reformulating a classical circle constraint as an explicit exchange relation, providing a transparent diagnostic framework for analyzing deviations from symmetric equilibrium in geometric and variance-partitioned systems.