Intriguing Invariants of Centers of Ellipse-Inscribed Triangles
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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 linear combination of barycenter and orthocenter, its locus is an ellipse; (ii) and that over the family of said linear 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.
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