Locus invariants of projected Poncelet triangles III: rigid rotations of generic caustic

Let T=ABC be a family of Poncelet triangles inscribed in a circle K=(O,R) and a circumscribing an ellipse E'. Let M={{m11,m12,0},{m21,m22,0},{m31,m32,1}} be a generic projectivity (no translation), and T'=A'B'C' be the image of T under M, i.e., T'=M * T. Note that the T' are also Ponceletian.
From [1] we know that the loci of any triangle center which is a fixed linear comb. of X2,X3, (e.g., X2, X3, X4, X5, ...) describe conics for any N=3 Poncelet family.
We show that over rigid rotations of E' about O, loci X(2), X(3), X(4) of T' conserve:
- X2, X4: eccentricity and axis orientation. When M is an affinity (m31=m32=0), then the X2 loci also conserve area.
- X3: if M is an affinity, area is conserved.
Note: see N=5 example [2]
geogebra: www.geogebra.org/classic/wbxr...
[1] Helman et al., "Poncelet triangles: a theory for locus ellipticity", Beiträge zur Algebra und Geometrie, v.63, 2019-2022. link.springer.com/article/10....
arXiv: arxiv.org/abs/2106.00715
[2] D. Reznik, "Locus Invariants of Projected Poncelet Bicentric Polygons VIII: vertex and area centroids", TH-cam, 2024. • Locus Invariants of Pr...