Poncelet triangles: conic loci of the orthocenter and of the isogonal conjugate of a fixed point

Abstract

We prove that over a Poncelet triangle family interscribed between two nested ellipses $\mathcal{E},\mathcal{E}_c$, (i) the locus of the orthocenter is not only a conic, but it is axis-aligned and homothetic to a $90^o$-rotated copy of $\mathcal{E}$, 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 $\mathcal{E}$). 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 $\mathcal{E}_c$ is a circle. The former case is the union of two circles!