Cevian operations on distributive lattices

We construct a completely normal bounded distributive lattice D in which for every pair (a, b) of elements, the set {x $\in$ D | a $\le$ b $\lor$ x} has a countable coinitial subset, such that D does not carry any binary operation - satisfying the identities x $\le$ y $\lor$(x-y),(x-y)$\land$(y-x) = 0, and x-z $\le$ (x-y)$\lor$(y-z). In particular, D is not a homomorphic image of the lattice of all finitely generated convex {\ell}-subgroups of any (not necessarily Abelian) {\ell}-group. It has $\aleph 2 elements. This solves negatively a few problems stated by Iberkleid, Mart{\'i}nez, and McGovern in 2011 and recently by the author. This work also serves as preparation for a forthcoming paper in which we prove that for any infinite cardinal$\lambda$, the class of Stone duals of spectra of all Abelian {\ell}-groups with order-unit is not closed under L$\infty$$\lambda$-elementary equivalence.Comment: 23 pages. v2 removes a redundancy from the definition of a Cevian operation in v1.In Theorem 5.12, Idc should be replaced by Csc (especially on the G side

Synthetic foundations of cevian geometry, I: Fixed points of affine maps in triangle geometry

We give synthetic proofs of many new results in triangle geometry, focusing especially on fixed points of certain affine maps which are defined in terms of the cevian triangle $DEF$ of a point $P$ with respect to a given triangle $ABC$, as well as the cevian triangle of the isotomic conjugate $P'$ of $P$ with respect to $ABC$. We prove a formula for the cyclocevian map in terms of the isotomic and isogonal maps using an entirely synthetic argument, and show that the complement $Q$ of the isotomic conjugate $P'$ has many interesting properties. If $T_P$ is the affine map taking $ABC$ to $DEF$, we show synthetically that $Q$ is the unique ordinary fixed point of $T_P$ when $P$ is any point not lying on the sides of triangle $ABC$, its anti-complementary triangle, or the Steiner circumellipse of $ABC$. We also show that $T_P(Q')=P$ if $Q'$ is the complement of $P$, and that the affine map $T_P T_{P'}$ is either a homothety or a translation which always has the $P$-ceva conjugate of $Q$ as a fixed point. Finally, we show that $P$ lies on the Steiner circumellipse if and only if $T_PT_{P'}=K^{-1}$, where $K$ is the complement map for $ABC$. This paper forms the foundation for several more papers to follow, in which the conic on the 5 points $A,B,C,P,Q$ is studied and its center is characterized as a fixed point of the map $\lambda=T_{P'} T_P^{-1}$.Comment: 25 pages, 6 figure

Synthetic foundations of cevian geometry, III: The generalized orthocenter

In this paper, the third in the series, we define the generalized orthocenter $H$ corresponding to a point $P$, with respect to triangle $ABC$, as the unique point for which the lines $HA, HB, HC$ are parallel, respectively, to $QD, QE, QF$, where $DEF$ is the cevian triangle of $P$ and $Q=K \circ \iota(P)$ is the $isotomcomplement$ of $P$, both with respect to $ABC$. We prove a generalized Feuerbach Theorem, and characterize the center $Z$ of the cevian conic $\mathcal{C}_P$, defined in Part II, as the center of the affine map $\Phi_P = T_P \circ K^{-1} \circ T_{P'} \circ K^{-1}$, where $T_P$ is the unique affine map for which $T_P(ABC)=DEF$; $T_{P'}$ is defined similarly for the isotomic conjugate $P'=\iota(P)$ of $P$; and $K$ is the complement map. The affine map $\Phi_P$ fixes $Z$ and takes the nine-point conic $\mathcal{N}_H$ for the quadrangle $ABCH$ (with respect to the line at infinity) to the inconic $\mathcal{I}$, defined to be the unique conic which is tangent to the sides of $ABC$ at the points $D, E, F$. The point $Z$ is therefore the point where the nine-point conic $\mathcal{N}_H$ and the inconic $\mathcal{I}$ touch. This theorem generalizes the usual Feuerbach theorem and holds in all cases where the point $P$ is not on a median, whether the conics involved are ellipses, parabolas, or hyperbolas, and also holds when $Z$ is an infinite point. We also determine the locus of points $P$ for which the generalized orthocenter $H$ coincides with a vertex of $ABC$; this locus turns out to be the union of three conics minus six points. All our proofs are synthetic, and combine affine and projective arguments.Comment: 34 pages, 7 figure

On Routh-Steiner Theorem and Generalizations

Following Coxeter we use barycentric coordinates in affine geometry to prove theorems on ratios of areas. In particular, we prove a version of Routh-Steiner theorem for parallelograms.Comment: 11 pages, 4 figure