12 problems

Let $Q_1,Q_2,Q_3\subseteq\mathbb{R}^2$ be three general real conics. A real circle is a circle defined over $\mathbb{R}$ that is tangent to each of $Q_1,Q_2,Q_3$. Maximality conjec…

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…

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 $\mathrm{GR}(21_4)$ be the Grünbaum–Rigby configuration, decomposed into subconfigurations $\mathrm S_{12}$, $\mathrm S_{13}$, and $\mathrm S_{23}$ with point orbits…

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…

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…

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…

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…

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…

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…

Ellipse conjecture. The set of all interior points $D$ of the triangle $ABC$ satisfying this condition is an ellipse.

Non-constructibility conjecture. If $x',y'\ne 0$, then one can choose the parameters $a$ and $b$ above so that the common point of the investigated conics does not admit a planar c…