Isogonal Conjugacy and Equal Focal Inverted Pedal Areas

Invert the vertices of a triangle about a point and form the pedal triangle of its inverse with respect to that same point. We derive its ordered signed area in terms of the original circumcircle power and the three vertex distances. The formula proves that every finite isogonal-conjugate pair off the sidelines produces equal, nonzero inverse-pedal areas. For a genuine three-periodic elliptic billiard, the optical property of the ellipse and the billiard reflection law make its two foci such a pair. This gives an elementary proof of invariant k908,b: the literal focal area quotient is one for every nondegenerate elliptic-caustic triangular orbit. Two exact orbits in the same ellipse and caustic show that the individual area can nevertheless vary. Classical triangle identities are explicitly separated from the application; no absolute novelty or priority is claimed. AMR-050-0069 (raw ID 5100069, frozen ulamai/UnsolvedMath v1.6.0): COMPLETE_PROOF of the exact N=3 invariant k908,b. Finite isogonal conjugates off triangle sidelines have equal nonzero ordered signed inverted-pedal areas. The actual foci of a genuine three-periodic billiard triangle with a nondegenerate confocal elliptic caustic satisfy those hypotheses, so the denominator is nonzero and the quotient equals 1 throughout the family. The equality was already stated as Observation 1 by Reznik, Garcia and Helman (arXiv:2012.03020v2, Section 3); this note supplies an explicit proof and nonvanishing argument, not a claim to discovering the observation. No other-period or neighboring-invariant closure is claimed. The PDF, LaTeX source and dependency-free exact Python checkers are included. This self-audited, AI-assisted preprint is unrefereed. No independent human review, formal proof-assistant verification or absolute priority is claimed. The bounded literature audit retains access gaps for the final subscription journal version and full book chapters.