52 problems

Alfaro's conjecture. If $A=B$, then a shortest $(A,B)$-network has no Steiner points and is a union of cycles.

Let $\Lambda_2$, $\Lambda_8$, and $\Lambda_{24}$ be the hexagonal lattice in $\mathbb{R}^2$, the $E_8$ root lattice in $\mathbb{R}^8$, and the Leech lattice in $\mathbb{R}^{24}$, r…

Unique-centre conjecture. If a simplex has a unique simplex centre, then it is equifacetal.

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…

Neoplatonic conjecture. Any $6$-net $\sigma$ has a realization, unique up to isometry, as an undented Euclidean polyhedron $\Delta(\sigma)$ built from equilateral triangles of side…

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…

Let $d$ be the metric induced on a finite subset of $\mathbb{R}^m$, let $\mathcal{I}$ be a subset of $\operatorname{peel}(d)$, and let $d|_{\mathcal{I}}$ denote the restriction of…

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 $\mathcal{S}\subseteq\mathbb{E}^n$ be a configuration. It is Ramsey if for every $r\in\mathbb{N}$ there exists a dimension $n=n(\mathcal{S},r)$ such that…

Let $M$ be a biharmonic submanifold of a Euclidean space. Chen's conjecture. Every such submanifold is minimal. Chen, Ishikawa, and Jiang proved the claim for biharmonic surfaces i…

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…

Centroid-preservation uniqueness conjecture. The centroid of $ABCD$ coincides with the centroid of $EFGH$ if and only if $X=X_2$.

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…

Let $\mathcal{T}_1$ be the construction field from the preceding discussion, and consider allowing general intersections with any of the transcendental curves under consideration.…

Consider the fields obtained by allowing general intersections with the transcendental curves mentioned in the source, namely trigonometric functions, the Quadratrix of Hippias, an…

Let $\mathcal{P}$ be the field created by allowing arbitrary partition of angles, and let $\mathcal{T}_1$ be the relevant construction field. The algebraic numbers in…

Let $\mathcal{T}_1$ be the field associated with the geometric construction system under consideration, and let $\mathcal{P}$ denote the field obtained by allowing arbitrary angle…

Continuous periodic tiling conjecture. Let $\Sigma$ be a bounded measurable subset of $\mathbb{R}^d$ of positive measure. Then the tiling equation

Centroid characterization conjecture. If $FGHI$ is a square independently of the choice of $E$, then the common center function must be the centroid.

Let $X\subset\mathbb{E}^d$ be a set, and call it non-spherical if it is not contained in the surface of a sphere of any dimension. Write $\ell_m$ for the relevant $m$-point line co…

Let $X\subset\mathbb{E}^d$ be a set. Call $X$ Ramsey if for every natural number $r$ there exists $n$ such that every $r$-colouring of $\mathbb{E}^n$ contains a monochromatic copy…

Let $ABCD$ be a general quadrilateral, and form the central quadrilateral from the four half triangles determined by its diagonals, using the same triangle center in each. Area-pre…

Let $ABCD$ be a general quadrilateral, and form the central quadrilateral by selecting one triangle center, specified by a single center function, in each of the four half triangle…

Let $ABCD$ be a tangential quadrilateral, and form the central quadrilateral by placing the chosen triangle center in the four component triangles determined by the diagonals. Cycl…