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
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.
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
Angus, why would you visit this problem other than if you were looking to cheat yourself!?
They're only asking for help
not to cheat
Really Melody. Why fake it?
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.
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