20 problems

Let a pair of ellipses admit Poncelet 3-periodics, and let $X_1$ denote the incenter of a member of the family. The pair is confocal when its two ellipses share the same foci. Ince…

Polar Harmonic-family conjecture. The sum of half-angle tangents is conserved for the polar image of the Harmonic family, for all $N>3$.

Stationary-orthocenter classification conjecture. A Poncelet triangle family maintains $X_4$ stationary only if the conics are configured as focal-$X_4$, MacBeath, or Dual.

Let $\mathcal{T}_{\circ}$ be the Poncelet family whose circumcircle is constrained by the circular caustic, and let $\Gamma$ denote the line associated with the circumcircle-incent…

Let $\mathcal{T}$ be the family of Poncelet triangles, and let $\mathcal{E}_c$ be the inner Poncelet caustic. Consider the envelope of the circumcircle as the triangles range over…

Let $\mathcal{E}$ be the outer ellipse and let $\mathcal{E}_c$ be the inner Poncelet caustic. Consider the two envelopes associated with the Poncelet family: the envelope of the ci…

Let $N>3$, and consider the polar image of the Harmonic family, a Poncelet family of $N$-gons. For each polygon, let $\theta_i$ denote its relevant angles, so that the half-angle t…

Let $X_4$ be the orthocenter of a Poncelet triangle family interscribed between two conics. The configurations called focal-$X_4$, MacBeath, and Dual are the three configurations i…

Let $\mathcal{L}_7$ and $\mathcal{L}_8$ denote the loci of the Gergonne point $X_7$ and Nagel point $X_8$, respectively, in a Poncelet triangle family. The pair of Poncelet conics…

Let $X_1$ be the incenter of a Poncelet triangle, and consider its locus as the triangle varies in a family interscribed between two conics. The pair is confocal when the two Ponce…

Let $X_7$ and $X_8$ be the Gergonne and Nagel points, respectively, of a Poncelet triangle family, and let their loci be denoted by $\mathcal{L}_7$ and $\mathcal{L}_8$. A Poncelet-…

Let a Poncelet triangle family be inscribed in a circle with inconic or caustic, and let $F$ vary over the circumcircle. For each $F$, let $W$ be the envelope point of the correspo…

Let a Poncelet triangle family be inscribed in a circle. Fix a point $F$ on the circumcircle, let $V$ be the perpendicular projection of $F$ onto the Simson line of a member of the…

Let a Poncelet triangle family be inscribed in a circle, and consider the directrices of its isogonal circumparabolas (CPs). Directrix-envelope parabola conjecture. Over any Poncel…

Let a pair of ellipses admit Poncelet 3-periodics, and consider the loci of the excenters of the triangles in the family. The pair is poristic when the associated Poncelet family h…

Let a pair of ellipses admit Poncelet 3-periodics, and let $X_1$ denote the incenter of a member of this family. If the pair is concentric at $O$, call $O$ the common center; other…

Let $\mathcal{X}$ be a triangle center, and consider its locus as the triangle ranges over Poncelet 3-periodics in a concentric pair of non-axis-aligned ellipses. Elliptic-locus ch…

Consider a non-concentric, non-axis-aligned pair of ellipses admitting Poncelet 3-periodics. For each 3-periodic, let its circumcircle and Euler circle be the circles associated wi…

Let a generic concentric pair consist of an outer ellipse and an inner ellipse admitting Poncelet 3-periodics. Let $X_3$ and $X_5$ denote the circumcenter and Euler center of a 3-p…

Let a pair of conics admit a family of Poncelet 3-periodics, and consider the locus traced by the incenter $X_1$ and by the excenters of the triangles in that family. Confocality c…