2025/08/04 by Garcia, Ronaldo A., Helman, Mark, Reznik, Dan
#51M04 #51N20 #51N35 #68T20 #FOS: Computer and information sciences #FOS: Mathematics #Graphics (cs.GR) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2508.02368
We prove that over a Poncelet triangle family interscribed between two nested ellipses E,Ec, (i) the locus of the orthocenter is not only a conic, but it is axis-aligned and homothetic to a 90o-rotated copy of E, and (ii) the locus of the isogonal conjugate of a fixed point P is also a conic (the expected degree was four); a parabola (resp. line) if P is on the (degree-four) envelope of the circumcircle (resp. on E). We also show that the envelope of both the circumcircle and radical axis of incircle and circumcircle contain a conic component if and only if Ec is a circle. The former case is the union of two circles!