[11] **viXra:1003.0272 [pdf]**
*replaced on 3 Apr 2010*

**Authors:** Florentin Smarandache

**Comments:** 12 pages

In this paper we review nine previous proposed and solved problems of elementary 2D
geometry [4] and [6], and we extend them either from triangles to polygons or polyhedrons,or
from circles to spheres (from 2D-space to 3D-space), and make some comments about them.

**Category:** Geometry

[10] **viXra:1003.0256 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 4 pages

In this article we present the two classical negations of Euclid's Fifth Postulate
(done by Lobachevski-Bolyai-Gauss, and respectively by Riemann), and in addition of
these we propose a partial negation (or a degree of negation) of an axiom in geometry.
The most important contribution of this article is the introduction of the degree of
negation (or partial negation) of an axiom and, more general, of a scientific or humanistic
proposition (theorem, lemma, etc.) in any field - which works somehow like the negation
in fuzzy logic (with a degree of truth, and a degree of falsehood) or like the negation in
neutrosophic logic [with a degree of truth, a degree of falsehood, and a degree of
neutrality (i.e. neither truth nor falsehood, but unknown, ambiguous, indeterminate)].

**Category:** Geometry

[9] **viXra:1003.0254 [pdf]**
*submitted on 26 Mar 2010*

**Authors:** Cătălin Barbu

**Comments:** 4 pages

In this note, we present a proof of Smarandache's cevian triangle
hyperbolic theorem in the Einstein relativistic velocity model of hyperbolic geometry.

**Category:** Geometry

[8] **viXra:1003.0245 [pdf]**
*submitted on 25 Mar 2010*

**Authors:** Cătălin Barbu

**Comments:** 4 pages

In this note, we present a proof of the hyperbolic a Smarandache's
pedal polygon theorem in the Poincaré disc model of hyperbolic geometry.

**Category:** Geometry

[7] **viXra:1003.0187 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Mihály Bencze, Florin Popovici, Florentin Smarandache

**Comments:** 5 pages

In this article we present a generalization of a Leibniz's theorem in geometry and
an application of this.

**Category:** Geometry

[6] **viXra:1003.0164 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 7 pages

In these paragraphs one presents three generalizations of the famous theorem of
Ceva

**Category:** Geometry

[5] **viXra:1003.0162 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 2 pages

In this short note we will prove a generalization of Joung's theorem in space.

**Category:** Geometry

[4] **viXra:1003.0116 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 23 pages

The goal of this paper is to experiment new math concepts
and theories, especially if they run counter to the classical
ones. To prove that contradiction is not a catastrophe, and
to learn to handle it in an (un)usual way.
To transform the apparently unscientific ideas into scientific
ones, and to develop their study (The Theory of Imperfections).
And finally, to interconnect opposite (and not only) human
fields of knowledge into as-heterogeneous-as-possible
another fields.

**Category:** Geometry

[3] **viXra:1003.0058 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Ion Pătraşcu

**Comments:** 5 pages, Translated by Prof. Florentin Smarandache

In this article we prove the theorems of the orthopole and we obtain, through
duality, its dual, and then some interesting specific examples of the dual of the theorem
of the orthopole.

**Category:** Geometry

[2] **viXra:1003.0057 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Ion Pătraşcu

**Comments:** 7 pages, Translated by Prof. Florentin Smarandache

The purpose of this article is to familiarize the reader with these notions, emphasizing on
connections between them.

**Category:** Geometry

[1] **viXra:1003.0056 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Ion Pătraşcu

**Comments:** 5 pages, Translated by Prof. Florentin Smarandache

In this article we elementarily prove some theorems on the poles and polars
theory, we present the transformation using duality and we apply this transformation to
obtain the dual theorem relative to the Samson's line.

**Category:** Geometry