Analytic solution for areas ratio

Find the ratio of the red area to the blue area. Regular heptagons are equal.

Inspiration1

Inspiration2 -Septic Triangle

Phyton give 2.

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from math import isclose
from sympy import *

radius = csc(pi / 7) / 2  # arrange for side length = 1
center = Point(0, 0)  # centered at Origin
p = Polygon(center, radius, n=7)
side = trigsimp(p.sides[0].length)
assert (isclose(side, 1, abs_tol=1e-30))

A, B, C, D, E, F, G = p.vertices
print(A)
print(B)
AE = Line(A, E)
GC = Line(G, C)
BF = Line(B, F)
GB = Line(G, B)
FA = Line(F, A)
P = FA.intersection(GB)[0]
H = AE.intersection(GC)[0]
J = AE.intersection(BF)[0]
I = GC.intersection(BF)[0]
area = Triangle(H, I, J).area*7
Big_area = Triangle(Point(0,0),A,B).area*7
Lit_area = Triangle(Point(0,0),J,I).area*7
SideL_area = Triangle(G,A,P).area*7
SideB_area = Triangle(G,A,H).area*7
S= Big_area-SideL_area-Lit_area-area
s= Big_area-SideB_area-Lit_area
print(S/s)
S_area = N(S, 20)
s_area =N(s,20)

print(S_area)
print(s_area)
print(S_area/s_area)


(-7*(-9 + cos(pi/14)/sin(pi/7) + 4*cos(3*pi/14)/sin(pi/7))/(32*(sin(pi/14) + 1)*sin(pi/14)*sin(pi/7)) - 7*(1 - sin(3*pi/14))*(-cos(3*pi/14) + cos(pi/14))/(4*(-sin(pi/7) + sin(2*pi/7) + cos(pi/14))*sin(pi/7)) - 7*(-cos(3*pi/14) + cos(pi/14))*(-789*cos(2*pi/7)/(8*sin(pi/7)) - 789*sin(3*pi/14)/(8*sin(pi/7)) - 385*sin(pi/14)/(4*sin(pi/7)) - 15*sin(5*pi/14)/(4*sin(pi/7)) - 15*(1 - cos(2*pi/7))**2*cos(2*pi/7)/sin(pi/7) - 23*(1 - cos(2*pi/7))**2*sin(pi/14)/sin(pi/7) - 3*(1 - cos(2*pi/7))**2*sin(3*pi/14)/sin(pi/7) + 9*(1 - cos(pi/7))**2/(2*sin(pi/7)) + 43*cos(3*pi/7)/(8*sin(pi/7)) + 2*(1 - cos(3*pi/7))**2/sin(pi/7) + 35*(1 - cos(2*pi/7))**2/(4*sin(pi/7)) + 10*(1 - cos(2*pi/7))**2/tan(pi/7) + 36/tan(pi/7) + 901/(8*sin(pi/7)))/(32*(sin(2*pi/7) + 2*sin(pi/7))*(-sin(pi/7) + sin(2*pi/7) + cos(pi/14))*(-6*cos(pi/14) + 5*sin(pi/7) + 4*cos(3*pi/14))*sin(pi/14)*sin(pi/7)**2) + 7*sin(2*pi/7)*csc(pi/7)**2/8)/(-7*(-48*sin(pi/7) - 22*cos(5*pi/14) - 7*sin(2*pi/7) + 35*cos(pi/14))/(4*(-60*cos(2*pi/7) - 49*sin(3*pi/14) - 75*sin(pi/14) - 20*sin(2*pi/7)**2 - 6*cos(pi/7) - 8*cos(2*pi/7)*cos(3*pi/7) + 8*cos(3*pi/7) + 38*sin(5*pi/14) + 66)) - 7*(1 - sin(3*pi/14))*(-cos(3*pi/14) + cos(pi/14))/(4*(-sin(pi/7) + sin(2*pi/7) + cos(pi/14))*sin(pi/7)) + 7*sin(2*pi/7)*csc(pi/7)**2/8)

2.4356342034533018176

1.2178171017266509088

2.0000000000000000000

#Geometry

Note by Yuriy Kazakov
3 months ago

No vote yet
1 vote

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Comments

As simple as that - we cut the big star into 21 pieces and make up two identical stars - one of the red and the second of the yellow and blue pieces.

Yuriy Kazakov - 3 months ago

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That's...incredible. Simply amazing Yuriy!

David Stiff - 3 months ago

Sympy! Cool!

David Stiff - 3 months ago

Yeah, sympy is fun to play with, isn't it? I wish it were a little better at simplifying trig expressions, though. That one is frightening me. :)

Fletcher Mattox - 3 months ago

Problem - find solution without Sympy. Only paper and pencil.

Yuriy Kazakov - 3 months ago

You should tweet this to Cshearer41 on Twitter with a caption of

"Prove that the area of the pink region is twice the area of the blue region"

Pi Han Goh - 3 months ago
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