Hexagonal Flanks, Confocal Parabolas, and a Focal Equilateral

The paper explicitly states and illustrates Proposition 2: the second isodynamic point X16 of the reference triangle T coincides with the X16 of the three flank triangles (and, in the extended tiling, with all flanks) . The candidate solution’s key step is invalid: it claims that inversion is conformal and therefore sends an equilateral triangle to an equilateral triangle (in the sense of the chord-triangle formed by the images of the vertices). This is false in general; inversion preserves angles between corresponding curves, but the straight-segment triangle formed by the images of the vertices is not obtained by mapping straight sides to straight sides, so equilateral shape is not preserved in this sense. The remainder of the candidate argument depends on this step and collapses. The core lemma the candidate uses (characterizing isodynamic points via inversion taking a triangle to an equilateral one) is classical and fine, but the misapplication of conformality renders the overall solution incorrect.