[5] **viXra:1205.0092 [pdf]**
*submitted on 2012-05-23 20:05:38*

**Authors:** Mircea Eugen Şelariu

**Comments:** 23 Pages.

Prezentarea ar trebui să începă cu funcţiile beta excentrice, deoarece ele vor fi
utilizate în continuare şi la definirea şi prezentarea următoarelor FSM-CE, care sunt
funcţiile amplitudine excentrică, funcţii asemănătoare din multe puncte de vedere cu
funcţiile eliptice Jacobi amplitudine sau amplitudinus am(u,k).
Dar va începe cu fucţia “rege” radial excentric rexθ şi Rexα.

**Category:** Geometry

[4] **viXra:1205.0060 [pdf]**
*submitted on 2012-05-13 16:00:45*

**Authors:** Hilário Fernandes de Araújo Júnior

**Comments:** 3 Pages.

The cosine's law shows that, if we have a triangle with sides a, b and c, and an angle α between the sides b and c, this relationship is right:
a²=b²+c²−2bc[cos α].Will be shown here this law deduction through the trigonometry's
fundamental relation.

**Category:** Geometry

[3] **viXra:1205.0055 [pdf]**
*submitted on 2012-05-11 20:15:25*

**Authors:** Hilário Fernandes de Araújo Júnior

**Comments:** 4 Pages.

In this article, is developed a π representation as an infinite sum, through a definite integral.

**Category:** Geometry

[2] **viXra:1205.0051 [pdf]**
*submitted on 2012-05-09 07:44:01*

**Authors:** Alberto Coe

**Comments:** 3 Pages.

Using elementary geometry we have performed an approach to Pi .this agrees to the fifth decimal place .

**Category:** Geometry

[1] **viXra:1205.0003 [pdf]**
*submitted on 2012-05-02 23:20:08*

**Authors:** Jay Yoon

**Comments:** 5 Pages.

I will present a proof of Euclid’s fifth postulate (I.Post.5) that proves, as an intermediate step, a proposition equivalent to it (I.32); namely, that in any triangle, the sum of the three interior angles of the triangle equals two right angles. The proof that I.32 implies I.Post.5 and vice versa is well-established and will be omitted for the sake of brevity. The proof technique is somewhat unorthodox in that it proves I.33, which states that straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel, before establishing I.32, contrary to the order in which the propositions are demonstrated in Euclid’s Elements.
Two triangle congruence theorems, namely the side-angle-side (I.4) and side-side-side congruence theorems (I.8) are employed in order to prove I.33 without recourse to I.Post.5 or any of its equivalent formulations. In addition, a parallelogram is constructed by an unorthodox method; namely, by defining the diagonals upon which the parallelogram’s sides will be determined prior to the sides themselves. The proof assumes the five common notions stated in Book I of The Elements without explicitly making a reference to them when they are used. Furthermore, a figure is presented with color-coded angles and sides, with angles of the same color being equal in measure and sides of both the same color and the same number of tick marks being equal in length. The sides *GH* and *EJ* enclosed by brackets are indicated to be equal in length, the reason for the different notation being that the tick marks were used in reference to the halves of *GH*, namely *OG* and *OH*. The tick marks then refer to the parts of *GH*, and the bracket refers to the whole of *GH*; the latter is then equated to *EJ* by I.33, which is proven before its use.

**Category:** Geometry