How to generate this---question of understanding

Thank you for this reply to my threat. I am quite puzzled now. For, if I'm not wrong, on Euclidean metric with n dimensions, $\displaystyle \lambda d(v,x) = (1 - \lambda) d(v,y) $ just in case $\displaystyle \Sigma_{1 \leq i \leq n} (\lambda v)^2 - (\lambda x)^2 = \Sigma_{1 \leq i \leq n} ((1 - \lambda)v)^2 - ((1 - \lambda)y)^2$. But the latter equation rewrites to a quadratic equation of the form: $\displaystyle \Sigma_{1 \leq i \leq n} (2\lambda - 1) x^2\ +\ ax\ +\ k = 0 $, which does not define a circle, if I am not wrong. It would be great if you could help me out of my state of puzzlement.