Geometry, Volume 1, Issue 1

It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths $a,b,$ and c of a triangle $ABC$. For example, the circumcenter is represented by the polynomial $a(b^{}$...

In this article, we provide elementary proofs of two projective generalizations of Hagge’s theorems. We describe Steiner’s correspondence as a projective generalization of inversion.

We describe a series of parallel lenses with constant proportions packed in a circle. To construct n lenses, a regular 2(n + 1)-gon is drawn with a central diagonal of 2r length, followed by an array of n parallel diagonals perpendicular to the forme...

Suppose that X is a triangle center with homogeneous coordinates (barycentric or trilinear) $x:y:z$. Eight unary operations discussed in this paper include $u_1\left(X\right)=(y-z)/x:(z-x)/y:(x-y)/z$. For each $u_i$, there exist, formally, two points,...

In the ancient realm of geometry, we have witnessed the ultimate display of mathematical abstract thought [...]