17 problems

A convex Poncelet polygon is a polygon in the projective plane whose vertices lie on one conic and whose edges are tangent to another conic; convexity means convexity in an affine…

Invariant side-square area conjecture. The sum of the areas of the constructed squares remains invariant throughout $\mathcal P$ if and only if the circumcenter of the triangle coi…

Let $\mathcal{E}$ and $\mathcal{E}_c$ be nested ellipses admitting a family $\mathcal{T}$ of Poncelet triangles, and let $T=ABC$ be a triangle in $\mathcal{T}$. Let $T'=A'B'C'$ be…

Area-invariance conjecture. The area $\mathrm{Area}(\triangle ABC)$ is independent of the choice of $\triangle ABC\in\mathcal{P}$ if and only if $\mathcal{C}$ and $\mathcal{D}$ are…

Let $\boldsymbol{P}^2(\mathbb{F}_q)$ be the projective plane over the finite field $\mathbb{F}_q$. For pencils $\mathscr{P}_{4}$ and $\mathscr{P}_{15}$, let $s$ range over…

Poncelet polygon density conjecture. The density satisfies

Let $\mathcal{H}$ be a Poncelet family of $N$-gons interscribed between two homothetic, concentric ellipses, and let $A_1$ and $A_2$ be the areas of the two lateral harmonic polygo…

Let $\cal{P}$ be a parabola, let $\cal{C}$ be a focus-centered circle, and let $\cal{R}$ be a Poncelet family of $N$-gons inscribed in $\cal{P}$ and circumscribed about $\cal{C}$.…

Let $\cal{P}$ be a parabola and $\cal{C}$ a focus-centered circle, and let $\cal{R}$ be a Poncelet family of $N$-gons inscribed in $\cal{P}$ and circumscribed about $\cal{C}$. Gene…

Let $\cal{P}$ be a parabola of focal distance $f$, and let $\cal{C}$ be a focus-centered circle of radius $r$ such that they admit a Poncelet family of inscribed pentagons. Pentago…

Let $N$ be a positive integer, and consider the families of parabola-inscribed Poncelet $N$-gons and their associated polar polygons, with the relevant centroids defined as in the…

Let $P_1P_2P_3$ be a bic-III family of triangles inscribed in an outer circle $\mathcal{C}$, with each side tangent to a distinct in-pencil circle, and assume all four in-pencil ci…

Let $P_1P_2P_3$ be a bic-III family of triangles inscribed in an outer circle $\mathcal{C}$, with each side tangent to a distinct in-pencil circle, and assume all four in-pencil ci…

Let $P_1P_2P_3$ be a bic-III family of triangles inscribed in an outer circle $\mathcal{C}$, with $P_1P_2$ and $P_1P_3$ tangent to distinct circles $\mathcal{C}'$ and…

Let $X$ be a triangle center that does not always lie on a triangle's circumcircle. Consider the bic-II family and its bic-I (poristic) subfamily. A locus is stationary when the co…

Conjectured values of $\tau_n$. Based upon experimental data,

Poncelet n-gon proportion conjecture. The proportion of conic pairs in $\mathbf P^2(\mathbb F_q)$ satisfying the Poncelet $n$-gon condition is asymptotically equal to