Two points $P_1$ and $P_2$ on the circumference of a circle (or on a sphere) are diametral, if the line segment $P_1P_2$ connecting them passes through the centre of the circle (resp. the sphere), i.e. is a diametre. Equivalently, the shortest distance of the diametral points $P_1$ and $P_2$ on the circle is maximal on the circle (resp. on the sphere), namely a half of the perimetre.

It's easily justified that a point of a circle (resp. a sphere) has exactly one diametral point.

A circle $c$ is a diametral circle of a given circle $c_0$ if $c$ intersects $c_0$ diametrically, i.e. in two diametral points of $c_0$

If the equation of $c_0$ is $(x-x_0)^2+(y-y_0)^2 = r^2$ , and $(a,\,b)$ , is inside $c_0$ then the equation of the diametral circle $c$ with centre $(a,\,b)$ , is given by $$(x-a)^2+(y-b)^2 = r^2-(x_0-a)^2-(y_0-b)^2.$$