Intriguing Invariants of Centers of Ellipse-Inscribed Triangles
This work addresses a specific geometric problem in mathematics, likely incremental as it builds on known triangle center theory and ellipse geometry.
The paper tackles the problem of characterizing invariants for centers of triangles inscribed in an ellipse with two fixed vertices, proving that certain affine combinations of barycenter and orthocenter yield elliptical loci and that these loci exhibit linear sweeping and translation properties.
We describe invariants of centers of ellipse-inscribed triangle families with two vertices fixed to the ellipse boundary and a third one which sweeps it. We prove that: (i) if a triangle center is a fixed affine combination of barycenter and orthocenter, its locus is an ellipse; (ii) and that over the family of said affine combinations, the centers of said loci sweep a line; (iii) over the family of parallel fixed vertices, said loci rigidly translate along a second line. Additionally, we study invariants of the envelope of elliptic loci over combinations of two fixed vertices on the ellipse.