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

TL;DR

This paper investigates the geometric loci related to Poncelet triangles, revealing that the orthocenter and isogonal conjugates of a fixed point form conics with specific properties, depending on the configuration of the nested ellipses.

Contribution

It establishes new geometric properties of the orthocenter and isogonal conjugates in Poncelet triangle configurations, including their conic nature and specific alignments.

Findings

Locus of the orthocenter is a conic, axis-aligned and homothetic to a rotated ellipse.

Locus of the isogonal conjugate of a fixed point is a conic, parabola, or line depending on the point's position.

Envelope of circumcircle and radical axis contains a conic component if and only if the inner conic is a circle.

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!