File:FS RCCC dia.png

Shows the largest circle within RCC sangaku.

Base is the right isosceles triangle of side length ${\displaystyle a}$ and centroid ${\displaystyle S_{0}}$
Inscribed is the largest possible circle with radius ${\displaystyle r_{1}}$ around point ${\displaystyle S_{1}}$
Inscribed is the largest possible circle outside the first circle with radius ${\displaystyle r_{2}}$ around point ${\displaystyle S_{2}}$
Inscribed is the largest possible circle outside the first two circles with radius ${\displaystyle r_{3}}$ around point ${\displaystyle S_{3}}$

0) The side length of the base right triangle ${\displaystyle |AB|=|AC|=a}$
1) Radius of the first circle ${\displaystyle r_{1}={\frac {2-{\sqrt {2}}}{2}}\cdot a\quad \approx 0,293\cdot a\quad }$ (See FS RC).
2) Radius of the second circle ${\displaystyle r_{2}=\left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}\cdot a\quad \approx 0.131\cdot a\quad }$, (See details under FS RCC dia.png)
3) For symmetry reasons is the radius of the third circle ${\displaystyle r_{3}=r_{2}=\left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}\cdot a\quad \approx 0.131\cdot a\quad }$

0) Perimeter of base triangle ${\displaystyle P_{0}=(2+{\sqrt {2}})\cdot a\quad \approx 3.41\cdot a\quad }$ (See FS R )
1) Perimeter of the first circle ${\displaystyle P_{1}=\pi \cdot (2-{\sqrt {2}})\cdot a\quad \approx 1.84\cdot a\quad }$ (See FS RC )
2) Perimeter of the second circle ${\displaystyle P_{2}=2\pi r_{2}=2\pi \left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}\cdot a\quad \approx 1.16\cdot a\quad }$ (See FS RCC )
3) Perimeter of the third circle is for symmetry reasons ${\displaystyle P_{3}=P_{2}=2\pi \left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}\cdot a\quad \approx 1.16\cdot a\quad }$

0) Area of the base triangle ${\displaystyle A_{0}={\frac {a^{2}}{2}}}$ (See FS R )
1) Area of the inscribed circle ${\displaystyle A_{1}=\pi \left({\frac {3-2{\sqrt {2}}}{2}}\right)\cdot a^{2}\quad \approx 0.27\cdot a^{2}\quad }$ (See FS RC)
2) Area of the second circle ${\displaystyle A_{2}=\pi r_{2}^{2}=\pi \left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{4}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{4}\cdot a^{2}\quad \approx 0.054\cdot a^{2}\quad }$(See FS RCC)
3) For the symmetry reasons is the area of the third circle ${\displaystyle A_{3}=A_{2}=\pi \left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{4}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{4}\cdot a^{2}\quad \approx 0.054\cdot a^{2}\quad }$

Centroid positions are measured from the centroid point of the base shape
0) Centroid positions of the base triangle: ${\displaystyle x_{0}=0\quad y_{0}=0}$
1) Centroid positions of the inscribed circle: ${\displaystyle x_{1}=y_{1}=\left({\frac {4-3{\sqrt {2}}}{6}}\right)\cdot a\quad \approx -0.04\cdot a\quad }$(See FS RC)
2) Centroid positions of the second circle:
${\displaystyle \quad x_{2}=\left({\frac {14}{3}}-{\frac {7}{2}}{\sqrt {2}}+(6-4{\sqrt {2}}){\sqrt {2+{\sqrt {2}}}}\right)\cdot a\quad \approx 0.351\cdot a\quad }$, (See FS RCC)
${\displaystyle \quad y_{2}=\left(\left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}-{\frac {1}{3}}\right)\cdot a\quad \approx -0.203\cdot a}$, (See FS RCC)
3) Centroid positions of the third are for symmetry reasons:
${\displaystyle \quad x_{3}=y_{2}=\left(\left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}-{\frac {1}{3}}\right)\cdot a\quad \approx -0.203\cdot a}$
${\displaystyle \quad y_{3}=x_{2}=\left({\frac {14}{3}}-{\frac {7}{2}}{\sqrt {2}}+(6-4{\sqrt {2}}){\sqrt {2+{\sqrt {2}}}}\right)\cdot a\quad \approx 0.351\cdot a\quad }$

In the normalised case the area of the base shape is set to 1.
${\displaystyle A_{0}=1\quad \Rightarrow {\frac {a^{2}}{2}}=1\quad \Rightarrow a={\sqrt {2}}}$

0) Side length of the base triangle ${\displaystyle a={\sqrt {2}}\quad \approx 1.414}$
1) Radius of the inscribed circle ${\displaystyle r_{1}={\frac {2-{\sqrt {2}}}{2}}\cdot {\sqrt {2}}={\frac {2{\sqrt {2}}-2}{2}}={\sqrt {2}}-1\quad \approx 0.414}$
2) Radius of the second inscribed circle ${\displaystyle r_{2}=\left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}\cdot {\sqrt {2}}\quad \approx 0.185}$
3) Radius of the third inscribed circle ${\displaystyle r_{3}=r_{2}=\left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}\cdot {\sqrt {2}}\quad \approx 0.185}$

0) Perimeter of base triangle ${\displaystyle P_{0}=(2+{\sqrt {2}})\cdot {\sqrt {2}}=2{\sqrt {2}}+2\quad \approx 4.8284271}$
1) Perimeter of the inscribed circle ${\displaystyle P_{1}=\pi \cdot (2-{\sqrt {2}})\cdot {\sqrt {2}}=\pi \cdot (2{\sqrt {2}}-2)\quad \approx 2.6025805}$
2) Perimeter of the second circle ${\displaystyle P_{2}=2\pi r_{2}=2\pi \left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}\cdot {\sqrt {2}}\quad \approx 1.1619551}$
3) Perimeter of the second circle ${\displaystyle P_{3}=P_{2}=2\pi \left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}\cdot {\sqrt {2}}\quad \approx 1.1619551}$

S) Sum of perimeters ${\displaystyle P_{s}=P_{0}+P_{1}+P_{2}+P_{3}\approx 9.7549179}$

0) Area of the base triangle ${\displaystyle A_{0}=1}$
1) Area of the inscribed circle ${\displaystyle A_{1}=\pi \left({\frac {3-2{\sqrt {2}}}{2}}\right)\cdot \left({\sqrt {2}}\right)^{2}=\pi (3-2{\sqrt {2}})\quad \approx 0.539}$
2) Area of the second inscribed circle ${\displaystyle A_{2}={\frac {\pi }{8}}\cdot \left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{4}\cdot \left(2-{\sqrt {2}}\right)^{4}\quad \approx 0.107}$
3) Area of the third inscribed circle ${\displaystyle A_{3}=A_{2}={\frac {\pi }{8}}\cdot \left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{4}\cdot \left(2-{\sqrt {2}}\right)^{4}\quad \approx 0.107}$

Centroid positions are measured from the centroid point of the base shape.
0) Centroid positions of the base triangle: ${\displaystyle S_{0}:\quad x_{0}=0\quad y_{0}=0}$
1) Centroid positions of the inscribed circle: ${\displaystyle S_{1}:x_{1}={\frac {2{\sqrt {2}}-3}{3}}=-0.057\quad y_{1}=x_{1}=-0.057}$
2) Centroid positions of the second circle:
${\displaystyle \quad x_{2}=\left({\frac {14}{3}}-{\frac {7}{2}}{\sqrt {2}}+(6-4{\sqrt {2}}){\sqrt {2+{\sqrt {2}}}}\right)\cdot {\sqrt {2}}=\left({\frac {14{\sqrt {2}}}{3}}-{\frac {7}{2}}\cdot 2+(6{\sqrt {2}}-4\cdot 2){\sqrt {2+{\sqrt {2}}}}\right)=\left({\frac {14{\sqrt {2}}}{3}}-7+2\cdot (3{\sqrt {2}}-4){\sqrt {2+{\sqrt {2}}}}\right)\quad \approx 0.496}$
${\displaystyle \quad y_{2}=\left(\left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}-{\frac {1}{3}}\right){\sqrt {2}}\quad \approx -0.286}$
3) Centroid positions of the third circle:
${\displaystyle \quad x_{3}=y_{2}=\left(\left(2-{\sqrt {2-{\sqrt {2}}}}\right)^{2}\cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}-{\frac {1}{3}}\right){\sqrt {2}}\quad \approx -0.286}$
${\displaystyle \quad y_{3}=x_{2}=\left({\frac {14}{3}}-{\frac {7}{2}}{\sqrt {2}}+(6-4{\sqrt {2}}){\sqrt {2+{\sqrt {2}}}}\right)\cdot {\sqrt {2}}=\left({\frac {14{\sqrt {2}}}{3}}-{\frac {7}{2}}\cdot 2+(6{\sqrt {2}}-4\cdot 2){\sqrt {2+{\sqrt {2}}}}\right)=\left({\frac {14{\sqrt {2}}}{3}}-7+2\cdot (3{\sqrt {2}}-4){\sqrt {2+{\sqrt {2}}}}\right)\quad \approx 0.496}$

01) The distance between the centroid of the base element and the centroid of the circle is:${\displaystyle d={\sqrt {(x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}}}=0.0808802...}$
02) The distance between the centroid of the base element and the centroid of the second circle is:${\displaystyle d={\sqrt {(x_{0}-x_{2})^{2}+(y_{0}-y_{2})^{2}}}=0.5730854...}$
03) The distance between the centroid of the base element and the centroid of the third circle is:${\displaystyle d={\sqrt {(x_{0}-x_{3})^{2}+(y_{0}-y_{3})^{2}}}=0.5730854...}$
12) The distance between the centroids of the first and the second circle is:${\displaystyle d={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}=0.5991445...}$
13) The distance between the centroids of the first and the third circle is:${\displaystyle d={\sqrt {(x_{1}-x_{3})^{2}+(y_{1}-y_{3})^{2}}}=0.5991445...}$
13) The distance between the centroids of the second and the third circle is:${\displaystyle d={\sqrt {(x_{2}-x_{3})^{2}+(y_{2}-y_{3})^{2}}}=1.1070746...}$

The sum of the distances is: ${\displaystyle D=3.5324145...}$

Apart of the base element there are three shapes allocated. Therefore the integer part of the identifying number is 3.
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.

${\displaystyle decimalpart(9.7549179...+3.5324145...)=decimalpart(13.2873324...)=.2873324}$

So the identifying number is: ${\displaystyle 3.2873324}$

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