Septic Triangle

Geometry Level 5

The figure shows a regular septagon with side length 1. △ H I J \triangle HIJ is formed by diagonals A E , B F AE, BF , and C G CG . What is the area of △ H I J \triangle HIJ ? Find a closed-form solution, convert it to a decimal number, and submit the sum of the first 50 digits to the right of the decimal point.


The answer is 239.

Fletcher Mattox by Fletcher Mattox , 72, USA

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2 solutions

Yuriy Kazakov
Mar 10, 2021

Find Area - use WolframAlpha

Find sum digits

Answer - 239.

0 Helpful 0 Interesting 1 Brilliant 0 Confused

How did you get x + y = 1 x+y=1 ?

Fletcher Mattox - 3 months ago

JI=y, IG=x, BJGA - rhombus. BJ=x+y=GA=1

Yuriy Kazakov - 3 months ago

Aha! Nice observation. Thank you.

Fletcher Mattox - 3 months ago
Fletcher Mattox
Mar 9, 2021 
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# Find the area of the smallest triangle in a fully connected unit heptagon
# https://www.geogebra.org/classic/rehfw2tx
# https://brilliant.org/problems/septic-triangle/

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-100))

A,B,C,D,E,F,G  = p.vertices
AE = Line(A, E)
GC = Line(G, C)
BF = Line(B, F)
H = AE.intersection(GC)[0]
J = AE.intersection(BF)[0]
I = GC.intersection(BF)[0]
area = Triangle(H, I, J).area
print(area)
N_area = N(area, 60)
print(N_area)
s_area = str(N_area)
digits = s_area[2:52]
digit_sum = sum(list(map(int, digits)))
print(digit_sum)


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(-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)
0.0191225580385579541933490594070239691892793864506960823994008
239

1 Helpful 1 Interesting 1 Brilliant 0 Confused

The area is 4 sin ⁡ 3 ( π 14 ) sin ⁡ ( π 7 ) \boxed{4 \sin^3\big(\tfrac{\pi}{14}\big) \sin\big(\tfrac{\pi}{7}\big)} and this is 0.0191225580385579541933490594070239691892793864506960823994007897861099600100983419092024586714785028599944956629719990953621880...

Peter Csorba - 3 months ago

Don´t you think that 50 digits is a bit too much when this is a geometry question. Because I got the correct answer, but my calculator could not give me 50 digits after decimal point.

Dan Czinege - 3 months ago

Yes, it was a bit immature of me. On the other hand, there is no excuse for letting precision stop you from answering the question assuming you have a closed-form expression. Just use Wolfram Alpha or any other online calculator.

Fletcher Mattox - 3 months ago

What do think about this - link text

Yuriy Kazakov - 3 months ago

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Nice! You are a man after my own heart.

Fletcher Mattox - 3 months ago

Well, I did not think of using Wolfram Alpha. At least I learned I can use it to calculate my answers with higher precision than calculator on my computer. :)

Dan Czinege - 3 months ago

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