finding $AX\cdot BX\cdot CX\cdot DX\cdot EX\cdot FX\cdot GX\cdot HX$

Question

Given a Regular Octagon $ABCDEFGH$, $AE=2$.

On $AE$ we choose point $X$ which dividing $AE$ in the ratio of $3:1$.

Need to find: $AX\cdot BX\cdot CX\cdot DX\cdot EX\cdot FX\cdot GX\cdot HX$

Any help?

Thanks.

EDIT: any method would be a good solution by me (complex numbers, vectors,...)

I tried to draw and got that $AX=\frac{3}{2}, XE=\frac{1}{2}$.

Tried using trig like like law of sines and analytic approach with coordinates but i'm stuck.

Answer 1

The monic polynomial $p$ whose roots are the numbers $\omega^k$ $\>(0\leq k\leq7)$ with $\omega:=e^{i\pi/4}$ is given by $$p(z)=\prod_{k=0}^7(z-\omega^k)=z^8-1\ ,$$ whereby the second representation stems from the fact that the $\omega^k$ are the $8^{\rm th}$ roots of $1$. It follows that $$\prod_{k=0}^7\left|\omega^k-{1\over2}\right|=1-{1\over 2^8}={255\over256}\ .$$

Answer 2

Hints: In the complex plane, your octagon can be assumed to be inscribed in the unit circle, with one vertex at $A = 1$. The remaining vertices $B, \dots, H$ are therefore eighth roots of unity.

Let $z$ be an arbitrary complex number (eventually $X$), and let $\omega$ denote a primitive eighth root; can you calculate the product $$ (z - A)(z - B) \cdots (z - H) = \prod_{j=0}^{7} (z - \omega^{j})? $$ Finally, the quantity you want is $$ |z - A| \cdot |z - B| \cdots |z - H| = \prod_{j=0}^{7} |z - \omega^{j}|. $$