I have a line going between two given points $\mathbf{S}$ and $\mathbf{E}$. I want to obtain the coordinates of the 6 points of the hexagon that lies in the plane perpendicular to the vector $(\mathbf{S} - \mathbf{E})$ that contains $\mathbf{S}$.
To solve this, I denoted the hexagon points $\mathbf{P}_k, k=1,\ldots, 6$ in the plane with $\mathbf{S}$ and noted that
$$ (\mathbf{P}_k - \mathbf{S})\cdot (\mathbf{E} - \mathbf{S}) = 0, \forall k. $$
I know that the norm $|\mathbf{P}_k - \mathbf{S}| = d, \forall k$ and that the norm of the line is $|\mathbf{S} - \mathbf{E}| = r$ and so using Pythagoras gives
$$ |\mathbf{P}_k - \mathbf{S}|^2 + |\mathbf{S} - \mathbf{E}|^2 = |\mathbf{P}_k - \mathbf{E}|^2 \iff \\ d^2 + r^2 = |\mathbf{P}_k - \mathbf{E}|^2, \forall k. $$
Moreover, I also know that if the origin of the coordinate system was $\mathbf{S}$ the points in the plane can be characterized as $$ (x_k, y_k, 0) = \left( d\cos\left(\frac{2\pi}{6}k\right), d\sin\left(\frac{2\pi}{6}k\right), 0\right), k=1,\ldots,6 $$
So far I obtained two out of three equations that I need to uniquely determine the points $\mathbf{P}_k$. My question is, how should I derive the third needed equation?