Triangle with qq identical lines

The task is: Given a triangle with qq identical lines, in this way as you can see on the picture. The orange lines are the same. Calculate γ\gamma!

ΔABC\Delta ABC' is an isosceles triangle, therefore β1=γ\beta_1=\gamma.

The sum of the angles at the B point is 180180^{\circ}, because AH is a line: 1802γ+β2=180\cancel{180^{\circ}}-2\gamma+\beta_2=\cancel{180^{\circ}} β2=2γ\beta_2=2\gamma

From the isosceles triangles and the sum of angles at the C point: β1+1802β2+β3=180    β3=4γγ=3γ\beta_1+\cancel{180^{\circ}}-2\beta_2+\beta_3=\cancel{180^{\circ}}\implies \beta_3=4\gamma-\gamma =3\gamma Same way β4=4γ,β5=5γ,,βn=nγ\beta_4=4\gamma, \beta_5=5\gamma, \cdots, \beta_n=n\gamma, because if we see an angle: βn2+1802βn1+βn=180    βn=2βn1βn2=(2(n1)n+2)γ=nγ\beta_{n-2}+\cancel{180^{\circ}}-2\beta_{n-1}+\beta_n=\cancel{180^{\circ}}\implies \beta_n=2\beta_{n-1}-\beta_{n-2}=(2(n-1)-n+2)\gamma=n\gamma This is true for the first, second and third angles, so this must be true for every angles.

If we have a triangle with qq lines, then the last β\beta is the q32\frac{q-3}{2}th.

Now in the bottom HGHHG'H' triangle: α+α+α+β6+β6=180\alpha+\alpha+\alpha+\beta_6+\beta_6=180^{\circ} or generally 3α+2βq32=1803\alpha+2\beta_{\frac{q-3}{2}}=180^{\circ}. But if we see the big triangle 2α+2βq32+γ=1802\alpha+2\beta_{\frac{q-3}{2}}+\gamma=180^{\circ}. Therefore α=γ\alpha=\gamma. After substituting: 3γ+2βq32=1803\gamma+2\beta_{\frac{q-3}{2}}=180^{\circ}. Replacing βq32\beta_{\frac{q-3}{2}}: 3γ+2q32γ=1803\gamma+2\cfrac{q-3}{2}\gamma=180^{\circ} 3γ+qγ3γ=1803\gamma+q\gamma-3\gamma=180^{\circ} qγ=180q\gamma=180^{\circ} γ=180q\boxed{\gamma=\cfrac{180^{\circ}}{q}}

Now can you solve this?

Note by Páll Márton
3 weeks, 2 days ago

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Nice proof! Explains where I went wrong! @Páll Márton.

Yajat Shamji - 3 weeks, 2 days ago

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Thank you! Did you solve my problem?

Páll Márton - 3 weeks, 2 days ago

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No! That's why I said it explains where I went wrong.

Yajat Shamji - 3 weeks, 2 days ago

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Angles, not angels.

Yajat Shamji - 3 weeks, 1 day ago

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Yeah. I use Word2016 to write the text and I just paste them there, but the default language isn't English so the autocorrector "corrects" my "typos".

Páll Márton - 3 weeks, 1 day ago

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I will add this word to the Word. Last time it wrote mooves instead of moves

Páll Márton - 3 weeks, 1 day ago

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Oh. I see. Doesn't happen with me though.

Also, tip:

If you wish to delete a problem, copy and paste the solution in Word, then copy the problem, go to create a new problem (open a new tab for this), paste the original problem's text, copy the title, paste it in the title section, change the answers (if needed) - if not, press Post. Then, copy the solution from Word, paste it and make any corrections (if needed) - if not, press Preview and Submit. Then close the Word Document.

Yajat Shamji - 3 weeks, 1 day ago

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@Yajat Shamji Or I will use txt

Páll Márton - 3 weeks, 1 day ago

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@Páll Márton Up to you.

I used this technique for my Hexadecimal Equations Problem 66 up to Version 44.

Yajat Shamji - 3 weeks, 1 day ago

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@Yajat Shamji this note isn't very famous I think you can continue this :)

Páll Márton - 3 weeks, 1 day ago

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@Páll Márton I will continue the prove chain. (albeit later)

Also, what if (for your next problem involving this):

1803600\frac{180}{3600}=0.05= 0.05 degrees

Yajat Shamji - 3 weeks, 1 day ago

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@Yajat Shamji I don't understand

Páll Márton - 3 weeks, 1 day ago

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@Páll Márton Triangle with 36003600 identical lines?

Yajat Shamji - 3 weeks, 1 day ago

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@Yajat Shamji Ohh. q is odd :)

Páll Márton - 3 weeks, 1 day ago

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@Páll Márton Else we can get only an intervallllll

Páll Márton - 3 weeks, 1 day ago

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@Páll Márton Triangle with 36013601 identical lines?

1803601\frac{180}{3601}=0.049986115= 0.049986115 degrees

Also, how do you do the degree sign? Baffles me...

P.S. Check out my note - Interesting Thing about Multiplication Last Digit Sequences (Even)

Yajat Shamji - 3 weeks, 1 day ago

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@Yajat Shamji Yeah. You just shifted the 2, 4, 6 and 8. ^{\circ}

Páll Márton - 3 weeks, 1 day ago

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@Páll Márton What about the triangle with 36013601 identical lines?

Yajat Shamji - 3 weeks, 1 day ago

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@Yajat Shamji I think with 45 lines was enaugh. If somebody can solve with 45, then he/she can solve with 3601 too.

Páll Márton - 3 weeks, 1 day ago

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@Páll Márton You mean enough, right?

Yajat Shamji - 3 weeks, 1 day ago

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@Yajat Shamji Yeah. This was my mistake :) English is too hard. To write from pronunciation is hard

Páll Márton - 3 weeks, 1 day ago

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Excellent exposition.

Pi Han Goh - 3 weeks, 1 day ago

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Thank you!

Páll Márton - 3 weeks, 1 day ago

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Wow! Good proof!

Vinayak Srivastava - 3 weeks ago

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Thank you!

Páll Márton - 3 weeks ago

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Excellent Proof @Páll Márton.

Aryan Sanghi - 1 week, 5 days ago

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Thank you!

Páll Márton - 1 week, 5 days ago

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