[26] **viXra:1607.0369 [pdf]**
*submitted on 2016-07-19 15:02:18*

**Authors:** Jeffrey Joseph Wolynski

**Comments:** 2 Pages. 3 illustrations

It is shown that simple geometry could have been used to make the discovery that planet formation is stellar evolution.

**Category:** Geometry

[25] **viXra:1607.0356 [pdf]**
*submitted on 2016-07-18 07:10:10*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 11 Pages.

In this article, we define the Lucas’s inner
circles and we highlight some of their properties.

**Category:** Geometry

[24] **viXra:1607.0355 [pdf]**
*submitted on 2016-07-18 07:11:14*

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

We present here the magic square of order n.

**Category:** Geometry

[23] **viXra:1607.0349 [pdf]**
*submitted on 2016-07-18 07:18:12*

**Authors:** Florentin Smarandache

**Comments:** 5 Pages.

Postulatul V al lui Euclid se enunta sub forma: daca o dreapta, care intersecteaza doua drepte, formeaza unghiuri interioare de aceeasi parte
mai mici decat doua unghiuri drepte, aceste drepte, prelungite la infinit, se intalnesc in parte a unde unghiurile interioare sunt mai mici decal doua unghiuri drepte.

**Category:** Geometry

[22] **viXra:1607.0348 [pdf]**
*submitted on 2016-07-18 07:18:58*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

In this article, we highlight some metric
properties in connection with Neuberg's circles and triangle. We recall some results that are necessary.

**Category:** Geometry

[21] **viXra:1607.0346 [pdf]**
*submitted on 2016-07-18 07:21:25*

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

It is a lot easier to deny the Euclid`s five postulates, than Hilbert`s twenty thorough axiom.

**Category:** Geometry

[20] **viXra:1607.0341 [pdf]**
*submitted on 2016-07-18 07:27:33*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

In 1969, fascinat de geometrie, am construit un spatiu partial eueliadian si partial neeuclidian
in acelasi timp, inlocuind postulatul V al lui Euclid (axioma paralelelor) prin urmatoarea
propozitie stranie continand cinci asertiuni.

**Category:** Geometry

[19] **viXra:1607.0331 [pdf]**
*submitted on 2016-07-18 07:38:11*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Les axes radicals de n cercles d'un même plan, pris deux à deux, dont les centres ne sont pas alignes, sont concourants.

**Category:** Geometry

[18] **viXra:1607.0330 [pdf]**
*submitted on 2016-07-18 07:40:58*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

In 1969, intrigued by geometry I constructed a partially euclidean and partially non-Euclidean space.

**Category:** Geometry

[17] **viXra:1607.0325 [pdf]**
*submitted on 2016-07-18 07:53:17*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 11 Pages.

In this article, we emphasize the radical axis of the Lemoine’s circles.

**Category:** Geometry

[16] **viXra:1607.0323 [pdf]**
*submitted on 2016-07-18 07:55:17*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

In this article, we define the first Droz-Farny’s circle, we establish a connection between it and a concyclicity theorem, then we generalize this theorem, leading to the generalization of Droz-Farny’s circle.

**Category:** Geometry

[15] **viXra:1607.0322 [pdf]**
*submitted on 2016-07-18 07:56:54*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

In this article, we prove the theorem
relative to the second Droz-Farny’s circle, and a sentence that generalizes it.

**Category:** Geometry

[14] **viXra:1607.0304 [pdf]**
*submitted on 2016-07-18 08:23:01*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 25 Pages.

In this article, we solve problems of geometric constructions only with the ruler, using known theorems.

**Category:** Geometry

[13] **viXra:1607.0303 [pdf]**
*submitted on 2016-07-18 08:23:58*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 6 Pages.

The late mathematician Cezar Cosnita, using the barycenter coordinates, proves two theorems which are the subject of this article.

**Category:** Geometry

[12] **viXra:1607.0302 [pdf]**
*submitted on 2016-07-18 08:24:44*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

In this article, we prove several theorems about the radical center and the radical circle of ex-inscribed circles of a triangle and calculate the radius of the circle from vectorial considerations.

**Category:** Geometry

[11] **viXra:1607.0260 [pdf]**
*submitted on 2016-07-18 05:32:50*

**Authors:** Florentin Smarandache

**Comments:** 14 Pages.

It is possible to de-formatize entirely Hilbert`s group of axioms of the Euclidean Geometry, and to construct a model such that none of this fixed axiom holds.

**Category:** Geometry

[10] **viXra:1607.0258 [pdf]**
*submitted on 2016-07-18 05:34:44*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

This article highlights some properties of
Apollonius’s circle of second rank in connection with the adjoint circles and the second Brocard’s triangle.

**Category:** Geometry

[9] **viXra:1607.0253 [pdf]**
*submitted on 2016-07-18 05:40:32*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

In this article, we prove the theorem relative to the circle of the 6 points and, requiring on this circle to have three other remarkable triangle’s points, we obtain the circle of 9 points (the Euler’s Circle).

**Category:** Geometry

[8] **viXra:1607.0240 [pdf]**
*submitted on 2016-07-18 06:07:02*

**Authors:** Florentin Smarandache

**Comments:** 5 Pages.

Let P, L be two sets, and r a relation included in PxL.

**Category:** Geometry

[7] **viXra:1607.0229 [pdf]**
*submitted on 2016-07-18 06:36:40*

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

Dans ces paragraphss on présente "trois généralisations du célèbre théorème de Ceva.

**Category:** Geometry

[6] **viXra:1607.0227 [pdf]**
*submitted on 2016-07-18 06:39:23*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Let’s consider the points A1,...,An situated on the same plane, and B1,...,Bn situated on another plane, such that the lines A1B1 are concurrent. Let’s prove that if the lines AiAj and BiBj are concurrent, then their intersecting points are collinear.

**Category:** Geometry

[5] **viXra:1607.0225 [pdf]**
*submitted on 2016-07-18 06:41:46*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

What happens in 3-space when the poiygon is replaced by a polyhedron?

**Category:** Geometry

[4] **viXra:1607.0209 [pdf]**
*submitted on 2016-07-18 07:05:18*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 7 Pages.

In this article, we get to Lemoine's circles
in a different manner than the known one.

**Category:** Geometry

[3] **viXra:1607.0208 [pdf]**
*submitted on 2016-07-18 07:06:38*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 8 Pages.

For the calculus of the first Lemoine’s circle, we will first prove.

**Category:** Geometry

[2] **viXra:1607.0166 [pdf]**
*replaced on 2016-07-26 09:29:46*

**Authors:** James A. Smith

**Comments:** Pages.

This document adds to the collection of solved problems presented in References 1-4. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the LLP limiting case of the Problem of Apollonius in three ways.

**Category:** Geometry

[1] **viXra:1607.0015 [pdf]**
*submitted on 2016-07-02 02:33:06*

**Authors:** Philip Gibbs

**Comments:** Pages. DOI: 10.13140/RG.2.2.13171.12325

Sixty years ago Richard Bellman issued a difficult challenge to his fellow mathematicians. If a rambler is lost in a forest of known shape and size, how can she find the best path to follow in order to escape as quickly as possible? So far solutions are only known for a handful of simple cases and the general problem has therefore been described as “unapproachable.” In this work a computational “random paths” method to search for optimal escape paths inside convex polygonal forests is described. In particular likely solutions covering all cases of isosceles triangles are given. Each conjectured solution provides a potentisl upper-bound for Moser’s worm problem. Surprisingly there are two cases of triangles which would provide improvements on the best known proven upper bounds.

**Category:** Geometry