43 problems

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 triangle vary through a family of 3-periodic orbits, and consider the Circumellipses in Moses' pencil, namely the Circumellipses whose centers lie on the Feuerbach Circumhype…

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…

Arc-partition conjecture. The whole orbit $\mathrm{LEB}(z)$ can be partitioned into a certain number of subsets, each lying on a certain arc that is tangent to the real axis at $0$…

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

Let $T$ be one of the target triangles in the incommensurable-angle cases considered in the paper, and let $R$ be the triangular tile. Macro-tile decomposition conjecture. The tria…

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…

Let $R=(5,8,7)$ be the triangular tile with the side of length $7$ facing the angle $\pi/3$, and let $T$ be an equilateral triangle. Minimality conjecture. The construction in the…

Let $T$ be a triangle and let $R$ be a tile satisfying the paper's conditions, with $N$ denoting the number of copies of $R$ in a tiling. Eventual classification conjecture. The co…

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 $X_1$ 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 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 $\triangle$ be a triangle with vertices $A$, $B$, and $C$. For a point $X$ in its plane, let $Q(X)$ be the triple of normalized stationary charges for which the gradient of the…

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…