13 problems

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

Let $f$ and $g$ be continuous functions on $[a,b]$, with $-\infty<a<b<+\infty$ and $f(x)<g(x)$ for all $x\transport?$. Let $\text{\boldmath Q}=(Q;z_1,z_2,z_3,z_4)$ be the quadrilat…

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 $ABCD$ 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…

Let $ABCD$ be a general quadrilateral, and let a triangle center be specified by a center function. The central quadrilateral is formed by placing the chosen center in the four com…

Let $ABCD$ be a general quadrilateral, and let a triangle center be specified by a center function in the angles of each component triangle. Two center functions are equivalent if…

Let $M>0$ be a positive real number. Consider exterior quadrilaterals with prescribed angles and vertices $A_1=1$, $A_2=0$, $A_3$, and $A_4$, and let their conformal modulus be $M$…

Quadrilateral conjecture. There should be another quadrilateral whose upper inclusions are each $AH+1$ and whose lower inclusions have index one larger.

Area inequality conjecture. For any such ellipse and quadrilateral,

The one-fifth area-ratio conjecture. The ratio satisfies $r=1/5$. This conjecture arises from a geometric construction in which the inner quadrilateral is formed from the vertices…