12 problems

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 $X_{99}$.

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

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

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 $X_k'$ denote a triangle center of the polar family, and suppose its locus is a circle with nonzero radius. Circular-locus pencil conjecture. That circle belongs to the parabol…

Let a tetrahedron be isodynamic, and let a center function assign a point on each face. A center function is hyperbolic when the cevians to the corresponding face centers form a hy…

Let a pair of conics admit a family of Poncelet 3-periodics, and consider the locus traced by the incenter $X_1$ and by the excenters of the triangles in that family. Confocality c…

Let $\mathcal{E}$ be an ellipse with foci $f_1$ and $f_2$, and let $P(t)$ sweep $\mathcal{E}$. For the triangle $\mathcal{T}_f(t)=f_1f_2P(t)$, denote by $X_k$ the $k$th triangle ce…