14 problems

Rigidity conjecture for circle maps of an oval
arXiv@arxiv
Let $\gamma$ be an oval. For directions $u$ and $v$ , let $f_u,f_v:\gamma\to\gamma$ be the involutions induced by the family of parallel lines in those directions, and set…
Conjectureopenconics differential-geometry dynamical-systems
0
Conjecture on invariant sums of squared side areas for cyclic polygons
arXiv@arxiv
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…
Conjectureopenconics euclidean-geometry poncelet-polygons
1
The area-invariance conjecture for concentric circular 3-Poncelet pairs
arXiv@arxiv
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…
Conjectureopenconics euclidean-geometry poncelet-polygons
1
Incidence conjecture for Klein's arrangements of lines and conics
arXiv@arxiv
Let $\mathrm{GR}(21_4)$ be the Grünbaum–Rigby configuration, decomposed into subconfigurations $\mathrm S_{12}$ , $\mathrm S_{13}$ , and $\mathrm S_{23}$ with point orbits…
Conjectureopenconfigurations conics incidence-geometry
1
Minthorn's conjecture on centers of conics through a quadrilateral's vertices
arXiv@arxiv
Let $Q$ be a quadrilateral, and consider all conics passing through its four vertices. Minthorn's conjecture. If $Q$ is convex, the set of centers of these conics is a hyperbola; i…
Conjecturesolvedaffine-geometry conics quadrilaterals
1
CodeParade's conjecture on centers of tangent conics to a quadrilateral
arXiv@arxiv
Let $Q$ be a quadrilateral, and consider all conics tangent to the four extended sides of $Q$ . CodeParade's conjecture. The set of their centers is a straight line. The conjecture…
Conjecturesolvedaffine-geometry conics quadrilaterals
1
Nonconic perimeter-centroid conjecture for polar polygons of bicentric images
arXiv@arxiv
Let $\cal{B}'$ be the conic-inscribed polar image of a generic bicentric family of $N$ -gons, and let $\cal{P}'$ be the conic to which $\cal{B}'$ is inscribed. Nonconic perimeter-ce…
Conjectureopenbicentric-polygons conics polarities
1
Perimeter-centroid conic conjecture for polar images of bicentric families
arXiv@arxiv
Let $\cal{B}'$ be the conic-inscribed polar image of a generic bicentric family of $N$ -gons with respect to its bicentric circumcircle. Perimeter-centroid conic conjecture. Over…
Conjectureopenbicentric-polygons centroids conics
2
Möbius-conjugacy rigidity conjecture for an oval
arXiv@arxiv
Let $\gamma$ be an oval. For points $P,Q$ , let $f_P,f_Q:\gamma\to\gamma$ be the involutions associated with the points, and set $F_{P,Q}=f_P\circ f_Q$ . Consider pairs for which eit…
Conjectureopenconics differential-geometry dynamical-systems
1
The ellipse conjecture for interior points associated with circumcenters
arXiv@arxiv
Ellipse conjecture. The set of all interior points $D$ of the triangle $ABC$ satisfying this condition is an ellipse.
Conjectureopenconics euclidean-geometry locus-problems
0
Serre's sharpness conjecture for rational conics in families
arXiv@arxiv
Serre's conjecture. The upper bounds for the number of members of such families having a rational point should be sharp; in particular, in the diagonal conic case the count should…
Conjectureopenarithmetic-geometry conics rational-points
0
Conjecture that only orders 7 and 11 yield tangent-free conic unions
arXiv@arxiv
Conjecture on exterior points and sets without tangents. The cases $q=7$ and $q=11$ are the only cases for which $\mathcal{C}\cup E$ is a set without tangents.
Conjectureopenconics finite-geometry projective-planes
0
Blokhuis–Seress–Wilbrink conjecture on exterior sets of a conic
arXiv@arxiv
Blokhuis–Seress–Wilbrink conjecture. If $q>31$ , there are no exterior sets consisting of $(q+1)/2$ non-collinear exterior points in $\mathrm{PG}(2,q)$ .
Conjectureopenconics finite-geometry projective-planes
1
Extension of the septagon conic ratio theorem to odd polygons
arXiv@arxiv
Odd-polygon extension conjecture. The Septagon conic ratio theorem extends to all odd polygons.
Conjectureopenconics projective-geometry
1