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
            
            
        Answers
                    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.
    
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.
    
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.
    
and yes I know what I just said.
                    Answered by
            Anonymous
            
    They're only asking for help
not to cheat
    
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
    
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?
    
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.
    
Also, STOP THE TOXICATION.
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