Consider an ellipsoid for the vector $\vec{x}$ with equation
\begin{equation} \sqrt{ \lVert\vec{x}\rVert^2+a^2}+\sqrt{\lVert\vec{x}+\vec{c}\rVert^2+a^2}=d \end{equation}
and parameters $a$, $d$ and $\vec{c}$.
Is it true, and how would you proove, the following statement:
The scalar product of $2\vec{x}+\vec{c}$ and $\vec{x}$ is always positive.
Specifically, $2\vec{x}+\vec{c}$ is the sum of the two vectors $\vec{x}$ and $\vec{x}+\vec{c}$ starting at the foci, ending and pointing to the point $\vec{x}$ on the ellipsoid.