Igor Minevich
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Graph Powers of Groups II: The RA Matrix Open
For a graph $Γ$ and group $G$, $G^Γ$ is the subgroup of $G^{|Γ|}$ generated by elements with $g$ in the coordinates corresponding to $v$ and its neighbors in $Γ$. There is a natural epimorphism $G^Γ\to (G/[G,G])^Γ$ with kernel $[G,G]^n \ca…
A quadrilateral half-turn theorem Open
If $ABC$ is a given triangle in the plane, $P$ is any point not on the extended sides of $ABC$ or its anticomplementary triangle, $Q$ is the complement of the isotomic conjugate of $P$ with respect to $ABC$, $DEF$ is the cevian triangle of…
View article: Graph Powers of Groups
Graph Powers of Groups Open
The Lights Out Puzzle, played on a graph $Γ$, has been studied using linear algebra over $\mathbb{F}_2$ and more generally over $\mathbb{Z}/k\mathbb{Z}$. We generalize the setting by allowing the states of vertices to be the elements of a …
View article: Parks: A Doubly Infinite Family of NP-Complete Puzzles and Generalizations of A002464
Parks: A Doubly Infinite Family of NP-Complete Puzzles and Generalizations of A002464 Open
The Parks Puzzle is a paper-and-pencil puzzle game that is classically played on a square grid with different colored regions (the parks). The player needs to place a certain number of "trees" in each row, column, and park such that none a…
Real elliptic curves and cevian geometry Open
We study the elliptic curve $E_a: (ax+1)y^2+(ax+1)(x-1)y+x^2-x=0$, which we call the geometric normal form of an elliptic curve. We show that any elliptic curve whose $j$-invariant is real is isomorphic to a curve $E_a$ in geometric normal…
Synthetic foundations of cevian geometry, IV: the TCC-perspector theorem Open
In this paper we give a completely synthetic proof of the TCC-perspector theorem, that the isogonal conjugate $γ(H)$ of the generalized orthocenter $H$ (defined in Part III of this series of papers), with respect to a triangle $ABC$ and a …
A cevian locus and the geometric construction of a special elliptic curve Open
In a previous paper we defined the circumconic of a triangle $ABC$ with respect to a point $P$ as the conic $\tilde C=T_{P'}^{-1}(N_{P'})$, where $N_{P'}$ is the $9$-point conic for the quadrangle $ABCP'$ with respect to the line at infini…
Vertex positions of the generalized orthocenter and a related elliptic curve Open
We study triangles $ABC$ and points $P$ for which the generalized orthocenter $H$ corresponding to $P$ coincides with a vertex $A,B$, or $C$. The set of all such points $P$ is a union of three ellipses minus $6$ points. In addition, if $T_…