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$ABCDEFG$ is a regular heptagon inscribed in a unit circle centered at $O$. $\ell$ is the line tangent to the circumcircle of $ABCDEFG$ at $A$, and $P$ is a point on $\ell$ such that triangle $AOP$ is isosceles. Let $p$ denote the value of $AP\cdot BP \cdot CP \cdot DP \cdot EP \cdot FP \cdot GP$. How do we determine the value of the value of $p^2$?

I have tried this problem and couldn't come up with a solution, but I did find a solution using complex numbers on the internet for this problem. If anyone can solve this without complex numbers I would appreciate it!

Link to complex number solution: http://www.artofproblemsolving.com/Wiki/index.php/Mock_AIME_1_Pre_2005_Problems/Problem_10

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    Complex numbers are really the natural setting for the solution to this problem. Without them, you'll need to deploy a lot of trigonometric identities. It's surely possible, but I think it won't be pretty.2012-03-10
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    Maybe you could work it out from $\prod_{k=0}^6\left((1-\cos\frac{2\pi k}{7})^2+(1-\sin\frac{2\pi k}{7})^2\right)$. But I agree with rahul. It's not just the complex numbers, but their relation to roots of polynomials that makes it so natural for this problem.2012-03-10

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