Inradii Among Us

The Equation exist for triangle

$\frac{r}{R}=cos{\alpha}+cos{\beta}+cos{\gamma}-1$

$r$ - inradius, $R$ - outradius.

Here 6 triangles with $R=1$ and sets of angles ${\alpha}_i,{\beta}_i,{\gamma}_i$

$1. \frac{\pi}{14}, \frac{6\pi}{14}, \frac{7\pi}{14}$

$2. \frac{\pi}{14}, \frac{5\pi}{14}, \frac{8\pi}{14}$

$3. \frac{\pi}{14}, \frac{4\pi}{14}, \frac{9\pi}{14}$

$4. \frac{\pi}{14}, \frac{3\pi}{14}, \frac{10\pi}{14}$

$5. \frac{\pi}{14}, \frac{2\pi}{14}, \frac{11\pi}{14}$

$6. \frac{\pi}{14}, \frac{\pi}{14}, \frac{12\pi}{14}$

$\sum_{i=1}^6 {S_{i} }={\pi}\sum_{i=1}^6{r_i}^2 = {\pi} \sum_{i=1}^6 (cos{\alpha_{i}}+cos{\beta_{i}}+cos{\gamma_{i}}-1)^2$

Use WolframeAlpha to calculate the sum - answer

$0.40765528133453....$

$56$

And another way with Python

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python

from sympy import *

radius = 1  # arrange for side length = 1
center = Point(0, 0)  # centered at Origin
p = Polygon(center, radius, n=14)
side = trigsimp(p.sides[0].length)
#assert (isclose(side, 1, abs_tol=1e-20))
def Scircle(A,B,C):
  S=Triangle(A,B,C).area
  a=A.distance(B)
  b=B.distance(C)
  c=C.distance(A)
  R=(2*S/(a+b+c))**2
  return R

A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14 = p.vertices
s1=N(Scircle(A8,A1,A2),18)
s2=N(Scircle(A8,A2,A3),18)
s3=N(Scircle(A8,A3,A4),18)
s4=N(Scircle(A8,A4,A5),18)
s5=N(Scircle(A8,A5,A6),18)
s6=N(Scircle(A8,A6,A7),18)
summa = N((s1+s2+s3+s4+s5+s6)*pi,18)
st_summa= str(summa)
digits =st_summa[2:16]
print(digits)
print(sum(list(map(int,digits))))

40765528133453
56