# Verification of focal inverted pedal areas for triangular billiards

The manuscript gives a complete analytic argument. For an ordered triangle
of signed area D and circumradius R, inversion about F followed by the
pedal construction has signed area

    I(F) = D p_F² / (4 R² |A-F|² |B-F|² |C-F|²),

where p_F is the power of F with respect to the original circumcircle.
The proof derives the inversion and signed pedal formulas with their cyclic
ordering. Isogonal barycentric coordinates give exactly the compensating
ratio of the squared vertex-distance products and squared circle powers.
This proves equal, nonzero areas for every finite isogonal pair off the
sidelines. The optical and reflection laws identify the actual ellipse
foci as such a pair for genuine three-periodic elliptic-caustic triangles.

The checker check_triangle_isogonal_area.py was actually run and its
originating output is retained as triangle-isogonal-root-rerun-1.json.
It returned PASS_EXACT_TRIANGLE_ISOGONAL_AND_SOURCE_N3_FIXTURE. It checks
400 rational isogonal pairs, the signed inverse-pedal formula, translation
and order reversal, unequal nonisogonal control areas, a generic
circumcircle-zero control that is not a source billiard, and rejection of
inversion at a vertex. Its exact Q(sqrt(55)) source fixture satisfies
the boundary, reflection and strict internal tangency equations, and has
common positive focal area 3sqrt(55)/6400.

The separately executed checker check_axis_triangles.py returned
PASS_EXACT_MULTQUADRATIC_CONTROLS. Its complete stdout is retained as
axis-triangles-actual-stdout.json. Two genuine three-periodic orbits of
x²/8+y²/5=1 share caustic parameter 40/9, with positive squared caustic
semiaxes 32/9 and 5/9. The boundary equations, normalized reflections,
internal contact parameters, distinct vertices and inversion/projection
denominators are checked exactly in Q(sqrt(2),sqrt(3),sqrt(5)). Their
paired focal areas are sqrt(2)/72 and 5sqrt(5)/576 respectively. This
disproves constancy of an individual area, not the source quotient.

Both portable checkers use only the Python standard library. Fraction
coefficients establish exact identities, and rational radical enclosures
establish signs where needed. Approximate displays do not decide any
assertion. Finite regressions corroborate the universal analytic proof;
the regression counts do not establish its universal quantifiers.

The result is COMPLETE_PROOF of source k908,b, N=3 only. Equality was
already stated in Observation 1 of the cited inversive-triangle paper;
classical ingredients are credited. The retained bounded literature audit
does not establish exhaustive novelty or absolute priority. This
self-audited, AI-assisted preprint is unrefereed, without independent human
review or formal proof-assistant certification.

The final English source compiled successfully with the native editor. The separately exported PDF has 4 pages, all actually rendered and visually inspected. The actual export log and its harmless inspected warnings are retained in the local artifact receipts; no warning-free claim is made.