32 problems

Euler-line concurrence conjecture. A center on the circumcircle is concurrent for the Euler line if and only if it lies in the orbit of $X_{99}$.

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…

Commensurable-sides conjecture. The tiling must have commensurable sides.

Let $ABCD$ be an ideal trapezoid, and let $(a,b,c)$ be a triple of pairwise coprime integers satisfying … Write $x$ and $y$ for the side parameters of the ideal trapezoid as in the…

Let $(a,b,c)$ be the side lengths of the triangular tile, with angles determined by $a,b,c$, and let $T$ be a triangle with angles $(2\alpha,2\beta,\alpha+\beta)$. Tile-count divis…

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 $X_4$ 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 a Sharygin triangle be a non-isosceles triangle whose bisectral triangle, formed by the intersections of its internal angle bisectors with the opposite sides, is isosceles. Sup…

Interior angle sum conjecture. The sum of the interior angles of any horizontal-like geodesic triangle is greater than $\pi$:

Let $\xi(D)=\lambda_2(D)/\lambda_1(D)$ be the fundamental ratio of the first two Dirichlet–Laplacian eigenvalues. Consider the moduli space of triangles after quotienting by the sy…

Let $T$ be a reference triangle, and let $T'$ be its $X_{55}$-reflection triangle. The six vertices of the P-hyperbolas of $T'$ passing through $X_{55}$ of $T$ are the six points a…

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 $P_1P_2P_3$ 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…

Let a triangle center be defined by barycentric coordinates that are rational functions of the squares of the triangle's sidelengths, and consider its locus over the relevant famil…

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…

A focus-inversive polygon is obtained from an $N$-periodic billiard polygon by the focus-inversive construction described in the paper. Its perimeter is denoted by $L^\dagger$. Per…

Let $\mathcal{E}$ be an ellipse with foci $f_1$ and $f_2$, and let $P(t)$ sweep $\mathcal{E}$. For the triangle $\mathcal{T}_f(t)=f_1f_2P(t)$, denote by $X_k$ the $k$th triangle ce…