User:Hans G. Oberlack/FS R

The rectangular isosceles triangle as base element.

General case

Segments in the general case

The side of the rectangular isosceles triangle ABC $|AB|=|AC|=s$
The segment BC is the hypothenusis h of the triangle. h depends on the length of the side:

Applying the Pythagorean theorem to the triangle ABC leads to:
$\quad |BC|^{2}=|AB|^{2}+|AC|^{2}=$
$\quad \Leftrightarrow |BC|^{2}=2\cdot s^{2}$
$\quad \Leftrightarrow |BC|={\sqrt {2}}\cdot s$

Perimeter in the general case

Perimeter of base rectangular isosceles triangle $P_{0}=|AB|+|BC|+|AC|=s+{\sqrt {2}}\cdot s+s=2\cdot s+{\sqrt {2}}\cdot s=(2+{\sqrt {2}})\cdot s$

Area in the general case

Area of the base rectangular isosceles triangle $A_{0}={\frac {s^{2}}{2}}$

Centroids in the general case

By definition the centroid points of a base shape are $x_{0}=0\quad y_{0}=0$. Relatively is the lower left point of the of the base of the triangle at: $x_{L}=-{\frac {s}{3}}\quad y_{L}=-{\frac {s}{3}}$

Normalised case

Area in the normalised case

In the normalised case the area of the base isosceles triangle is set to $A_{0}=1$.

Segment in the normalised case

With $A_{0}={\frac {s^{2}}{2}}=1\quad \Rightarrow s^{2}=2\quad \Rightarrow s={\sqrt {2}}=1.41421...$

Perimeter in the normalised case

Perimeter of base rectangular isosceles triangle $P_{0}=(2+{\sqrt {2}})\cdot s=(2+{\sqrt {2}})\cdot {\sqrt {2}}=2\cdot ({\sqrt {2}}+1)=4.8284271...$

Centroids in the normalised case

The positions of lower left point of the base rectangular isosceles triangle:
$\quad x_{L}=-{\frac {s}{3}}=-{\frac {\sqrt {2}}{3}}=-0.4714...$
$\quad y_{L}=-{\frac {s}{3}}=-{\frac {\sqrt {2}}{3}}=-0.4714...$

Identifying number

Apart of the base element there is no other shape allocated. Therefore the integer part of the identifying number is 0.
The decimal part of the identifying number is the decimal part of the sum of the perimeters and the distances of the centroids in the normalised case.

$decimalpart(4.8284271...+0)=decimalpart(4.8284271...)=.8284271...$

So the identifying number is: $0.8284271$