$ABCDEFG$ 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