Residual Associate Triangles of General Inscribed and Pedal Configurations: Universal Single-Triangle Area Laws, the 2/9 Simson–Wallace Invariance, and a Resolution of the Kiss–András Orthology Problem

For an arbitrary triangle A1B1C1 inscribed in ABC we extend the theory of residual (corner) associate triangles. A coordinate-free edge-vector identity yields an exact single-triangle area formula in terms of the three residual area ratios, recovering the András–Kiss theorem on residual centroids, and shows that the associate triangle is perspective with ABC exactly when A1B1C1 is Cevian. For the pedal triangle of a point M, the triangle of residual centroids has area (9R^2 - OM^2)/(36R^2)[ABC]; this gives a 2/9 area invariance for M on the circumcircle and a collinearity circle of radius 3R. We determine when the associate triangle is orthologic to ABC, and for the orthic residual triangles prove that the triangle of residual X(n)-centers is orthologic to ABC (reciprocal center the isogonal conjugate of X(n)) and orthologic to the orthic triangle exactly when X(n) lies on the Euler line. Preprint of a manuscript submitted for publication (submitted version, 5 November 2026).