Asked by Anoymous

A regular dodecagon P_{1} P_{2}P_{3}...P_{12} is inscribed in a circle with radius $1.$ Compute \[(P_1 P_2)^2 + (P_1 P_3)^2 + \dots + (P_{11} P_{12})^2.\](The sum includes all terms of the form $(P_i P_j)^2,$ where $1 \le i < j \le 12.$) sorry if it's in latex
Answered by oobleck
so why use LaTex? You could always just type it in.
There are n(n-3)/2 diagonals in an n-gon.
So now you just need to get the lengths of the diagonals.
This site should help with that.

https://rechneronline.de/pi/decagon.php

Come back if you get stuck. I suspect that starting with a square, increasing the number of sides will show a pattern.
Answered by Anoymous
Still stuck :(
Answered by reeeeeee
stop cheating kid
Answered by Melody
This is not. how we help
Answered by Nacerima
Really Melody. Why fake it?
Answered by Jeff
CHEATER! STOP CHEATING! CHEATER!!!!
Answered by Bob
@Jeff, You're right. Anoymous is cheating. Why are you cheating? There's no point in cheating.
Answered by Larry
You realise that AoPS can ban your I.P. from its website, for academic dishonesty.
Answered by Melody
This is not. how we help. This is not. how we help. This is not. how we help.
Answered by Steve
Stop it cheater! That's an AoPS problem!
Answered by Angus
Why do you want to cheat? Do you care so much about the coloured bar on your progress chart that you do this!?
Answered by Melody
Angus, why would you visit this problem other than if you were looking to cheat yourself!?
Answered by dont cheat pls!
Cheating is bad!
Answered by Anonymous #2
First of all, tracking someone's I.P. address just because they asked a question on a math website is illegal.
Second of all, this person is just trying to get help on a problem they don't know. Geez. And half of you are only on here because you looked the problem up as well. Talk about hypocrisy.
Third, stop trolling people.
Answered by HELPFUL
For all you AoPS people who are crying on their tables, pleading for the right answer, because you are too lazy and cannot get the initiative, the answer is 144
Answered by Somebody
Okay seriously. You guys being on here means that you would have FOUND this problem and it wouldn't be easy unless you TYPED the problem in the search bard so you could have cheated as well!!

and yes I know what I just said.
Answered by Anonymous
They're only asking for help
not to cheat
Answered by yeet
brainly answer

Let denote the vector starting at the origin and ending at the vertex of the 12-gon. There is an angle of (360/12)º = 30º between consecutive vectors.

Recall that for any two vectors , we have



with the angle between the two vectors. Also recall that



For , is the length of the vector . So



The 12-gon is inscribed in a circle of radius 1, which means each vector has length 1, and from this we have



where is the angle between vectors and with , and these angles are multiples of 30º.

There are terms in the sum (from 12 total vertices, you take 2 at a time).

11 of these terms are the squared distances between consecutive vertices and separated by 30º, equal to ;
10 of them are the squared distances between vertices that are two vertices apart, separated by 60º, equal to ;
9 of them are the squared distances between vertices that are three vertices apart, separated by 90º, equal to ;
and so on, down to the 1 remaining uncounted squared distance between vertices that are ten vertices apart, separated by 330º, .
So we have



Answered by you_look_like_an_ogre
cheet alert cheet alert cheet alert
Answered by darthMat
Yall this was my friends problem and I had no idea how to help so I searched this up. Lol guys you found this because you searched it up. This was not my homework, just tryna help a friend. I could not have found this if I didn't search it up. Talk about being hypocrites. Also I just wanted help not some toxic behavior. Lastly tracking IP addresses on another website is illegal.
Answered by jlksdfj
yo @helpful lmao thx
Answered by Ariana(KayKay)
Wowowo! Let's not get into a battle here!

btw... can you explain it @HELPFUL?
Answered by pls_stop_being_toxic
Even when people get the answers, if you explain how to find the answer people CAN LEARN even if it's considered cheating. Then they will be able to solve a similar problem on their own next time. Even if they cheat without learning anything or just get the answer, it doesn't concern you so stop acting toxic and calling people cheaters. If they are cheaters let them cheat, karma will come for them, you don't have to do karma's job.
Answered by STOP THE TOXICATION
My hint for those who want explanation is that you can find a few right triangles in side the dodecagon if you add some lines. Also, I can confirm that Helpfuls answer is correct.

Also, STOP THE TOXICATION.
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