Close pair among three equidistant points of a thin annulus, all radii

Let $c,p\in\mathbb R^2$, let $q_0,q_1,q_2\in\mathbb R^2$, and let $\rho,x,\sigma,t$ be real numbers with $\rho>0$, $\sigma>0$ and

Assume that $\rho^2(1-x)\le |p-c|^2$, that each $q_i$ lies in the closed annulus

and that $q_0,q_1,q_2$ are equidistant from $p$, that is $|p-q_i|=|p-q_j|$ for all $i,j$. Then there are indices $i\ne j$ with

No bound on the common distance $|p-q_i|$ is assumed, and for $p$ only the lower annulus bound is assumed. The points $q_i$ need not be distinct.

For small $x$ the admissible threshold is close to $t=x(1+1/\sigma)\approx 2x$: three points of a thin annulus on any circle about such a point $p$ contain a pair at squared distance about $2x\rho^2$. It is the all-radius counterpart of the close-pair statement for circles of radius at most $\sqrt3\,\rho$.

Formalization Note Points are elements of EuclideanSpace ℝ (Fin 2), the triple is a function Fin 3 → ℝ², and all thresholds are stated for squared distances.