Conics Associated with a Cevian Nest

Synthetic Cevian Geometry Synthetic Cevian Geometry 1

Cubics Associated with Triangles of Equal Areas

The locus of a point X for which the cevian triangle of X and that of its isogonal conjugate have equal areas is a cubic that passes through the 1st and 2nd Brocard points. Generalizing from isogonal conjugate to P -isoconjugate yields a cubic Z(U,P ) passing through U ; if X is on Z(U, P ) then the P isoconjugate of X is on Z(U,P ) and this point is collinear with X and U . A generalized equal...

Heron Triangles: A Gergonne-Cevian-and-Median Perspective

We give effective constructions of Heron triangles by considering the intersection of a median and a cevian through the Gergonne point.

Applications of the Cevian Group in a Triangle

We first define and develop the geometric properties of the Cevian group in a triangle. This Abelian group distributes over the plane Euclidean projective space. We also develop the properties of the basic, the isotomic and the isogonal conjugates of a point in a triangle. We later proceed to transfer four points on the Euler line to the harmonic axis of the centroid of a triangle. Then using t...

On nest modules of matrices over division rings

Let $ m , n in mathbb{N}$, $D$ be a division ring, and $M_{m times n}(D)$ denote the bimodule of all $m times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m times n}(D)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $D$. Next, we introduce the notion of a nest module of matrices with entries from $D$. We ...