[3] **viXra:1311.0141 [pdf]**
*replaced on 2014-01-13 19:53:33*

**Authors:** Morio Kikuchi

**Comments:** 11 Pages.

We generalize inversion.

**Category:** Geometry

[2] **viXra:1311.0115 [pdf]**
*replaced on 2013-11-28 13:07:36*

**Authors:** Nathan O. Schmidt

**Comments:** 16 pages, 4 figures, accepted in Algebras, Groups and Geometries

In this work, we deploy Santilli's iso-dual iso-topic lifting and Inopin's holographic ring (IHR) topology as a platform to introduce and assemble a tesseract from two inter-locking, iso-morphic, iso-dual cubes in Euclidean triplex space. For this, we prove that such an "iso-dual tesseract" can be constructed by following a procedure of simple, flexible, topologically-preserving instructions. Moreover, these novel results are significant because the tesseract's state and structure are directly inferred from the one initial cube (rather than two distinct cubes), which identifies a new iso-geometrical inter-connection between Santilli's exterior and interior dynamical systems.

**Category:** Geometry

[1] **viXra:1311.0038 [pdf]**
*submitted on 2013-11-06 00:56:16*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 112 Pages.

This book contains 21 papers of plane geometry.
It deals with various topics, such as: quasi-isogonal cevians,
nedians, polar of a point with respect to a circle, anti-bisector,
aalsonti-symmedian, anti-height and their isogonal.
A nedian is a line segment that has its origin in a triangle’s vertex
and divides the opposite side in n equal segments.
The papers also study distances between remarkable points in the
2D-geometry, the circumscribed octagon and the inscribable octagon,
the circles adjointly ex-inscribed associated to a triangle, and several
classical results such as: Carnot circles, Euler’s line, Desargues
theorem, Sondat’s theorem, Dergiades theorem, Stevanovic’s
theorem, Pantazi’s theorem, and Newton’s theorem.
Special attention is given in this book to orthological triangles, biorthological
triangles, ortho-homological triangles, and trihomological
triangles.
Each paper is independent of the others. Yet, papers on the same or similar
topics are listed together one after the other.
The book is intended for College and University students and instructors that
prepare for mathematical competitions such as National and International
Mathematical Olympiads, or for the AMATYC (American Mathematical
Association for Two Year Colleges) student competition, Putnam competition,
Gheorghe Ţiţeica Romanian competition, and so on.
The book is also useful for geometrical researchers.

**Category:** Geometry